Electromagnetic drive double-shaft nanometer positioning platform and design method thereof

Abstract

The invention discloses an electromagnetic drive double-shaft nanometer positioning platform and a design method thereof. The platform comprises a flexible constraint mechanism and a driving main body mechanism. The flexible restraining mechanism comprises a cover plate and a flexible mechanism installed in the cover plate. The driving main body mechanism comprises a stator, a permanent magnet, an exciting coil and an armature; the armature is located in the center of the stator. And the four permanent magnets and the four exciting coils are sequentially and alternately arranged into a ring around the armature. And the armature is fixed with a central moving platform in the flexible mechanism. Through the guide mechanism, the platform realizes good flexibility, high precision and controllable motion trail. The platform adopts the Maxwell force principle for electromagnetic driving, shows wide driving capacity, and has remarkable advantages in hundred-micron multi-axis driving due to the unique characteristics of non-contact type, high force density and adjustable stroke. A bidirectional progressive structure topological optimization method is adopted in the flexible mechanism design, the structure is novel, and the difficulty is low when multiple performance indexes are calculated.

Description

Technical Field

[0001] The invention belongs to the technical field of compliant mechanism topology optimization of a nano-positioning platform, and particularly relates to an electromagnetic-driven biaxial nano-positioning platform and a design method thereof. Background Art

[0002] In the field of precision engineering, nano-positioning systems have always been a research hotspot. Especially in the context of the continuous development of fields such as micro / nano manufacturing, precision instruments, and biomedical engineering, the requirements for positioning accuracy and working stroke of related equipment are becoming increasingly stringent. According to the driving force principle, precision drive technologies are mainly divided into: electromagnetic drive and non-electromagnetic drive.

[0003] For electromagnetic force drive with nanometer precision requirements, the voice coil motor is the most common electromagnetic drive method. Its working principle mainly generates the Lorentz electromagnetic force perpendicular to the magnetic field direction and the current direction by the coil cutting the magnetic induction line as its main power source, and it essentially belongs to the driving force of the shear stress principle. Since the Lorentz force actuator has a simple working principle, can achieve a large working stroke within a certain frequency range, and does not require an intermediate transmission mechanism, it is often used in the nano-positioning platform system. However, due to its low force density, its response speed is relatively slow, which makes the Lorentz force drive method mainly used for large-stroke drives of millimeters or tens of millimeters. Compared with the voice coil motor, the piezoelectric actuator, as a non-electromagnetic drive method, has the advantages of a compact structure, a large driving force, and a fast response speed, and is widely used in the field of tool servo nano-positioning platforms. However, its micro-displacement output is extremely limited, usually within the range of dozens of micrometers. Usually, in order to meet the motion stroke requirements, the micro-displacement output usually needs to be designed into a cascaded displacement amplification mechanism, such as a lever mechanism, a bridge mechanism, and a Scott–Russell mechanism, etc., which can effectively increase its output stroke. However, the cascaded structure of the flexible hinge not only increases the complexity of the mechanism but also increases the moving mass of the system, thereby reducing the response speed of the system and severely limiting its frequency response range. Therefore, non-electromagnetic technologies are usually applied to small-stroke fields, such as micro / nano cutting and high-speed scanning of atomic force microscopes. Different from the traditional voice coil motor drive, the electromagnetic drive based on the Maxwell force actuator generates a driving magnetic field and a bias magnetic field on the excitation coil and the permanent magnet on both sides of the mover respectively, and uses 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 here, and it essentially belongs to the driving force of the normal stress principle. The Maxwell force drive has a wider driving ability. The same structural form can not only achieve small-stroke, high-bandwidth drive similar to the piezoelectric actuator but also independently achieve millimeter-level large-stroke, low-bandwidth drive similar to the voice coil motor. From the above analysis, it can be seen that both the piezoelectric and voice coil motor drives have principled disadvantages and it is difficult to achieve high-performance multi-axis drive at a stroke of hundreds of micrometers. Although the Maxwell force electromagnetic drive has been less studied in micro-nano drive, its unique properties such as non-contact, high force density, and flexible stroke adjustment make it have outstanding advantages in multi-axis drive of hundreds of micrometers, and also provide a new solution for the realization of high-performance multi-axis micro-nano drive.

[0004] The large stroke and high-frequency response of a nano-positioning platform system are often conflicting issues. Therefore, in practical applications, to prioritize ensuring the stroke of the nano-positioning system, the high-frequency tracking ability usually has to be sacrificed. To solve the above problems, there is an urgent need to propose an optimized design method. Currently, the design schemes of flexible mechanisms mainly develop along two directions: the pseudo-rigid body method and the topology optimization method. The former aims to separately achieve type synthesis and scale synthesis, while the latter comprehensively considers type synthesis and scale synthesis. The basic concept of the pseudo-rigid body method is to replace the traditional connection 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 has been widely used in the mechanism design in the field of precision positioning and operation. However, the defect of this method is that it must start from a rigid link mechanism, which makes the design highly dependent on the designer's experience. The basic idea of the topology optimization method is to seek the best distribution of materials within a given design domain to achieve the optimum of specific performance.

[0005] Currently, the topology optimization of flexible mechanisms has been widely applied in various fields, such as aerospace, energy systems, medical devices, and civil engineering, etc. A variety of methods have been comprehensively studied and developed, including the variable density method, the evolutionary structural optimization method, and the level set method, etc. Among them, the bi-directional evolutionary structural optimization method has been widely welcomed due to its high computational efficiency, algorithm robustness, simplicity, and ease of implementation. Summary of the Invention

[0006] The purpose of the present invention is to provide an electromagnetic-driven biaxial nano-positioning platform and its design method; this design method is based on the bi-directional evolutionary structural optimization method to design the flexible structure of the relevant biaxial nano-positioning platform, proposes a new method for optimized synthesis based on a double-layer flexible parallel mechanism, and adopts a multi-objective optimization design formula of the weighted sum of frequency and stiffness. This design method not only considers various factors affecting the design of the nano-positioning platform, but also regards the flexible structure sub-chain as the design domain of the topology optimization problem. The present invention realizes high-precision two-degree-of-freedom motion control, improves the accuracy and stability of the positioning system during the motion process, and brings significant technological progress and application prospects to the relevant fields.

[0007] In the first aspect, the present invention provides an electromagnetic-driven biaxial nano-positioning platform, which includes a flexible constraint mechanism and a driving main body mechanism. The flexible constraint mechanism includes a cover plate and a flexible mechanism installed inside the cover plate. The flexible mechanism includes a central moving platform, and four flexible connection sub-chains distributed around the central moving platform; both ends of the flexible connection sub-chains are respectively fixed to the central moving platform and the cover plate.

[0008] The driving main body mechanism includes a stator, a permanent magnet, an exciting coil, and an armature; the armature is located at the central position of the stator. Four permanent magnets and four exciting coils are both fixed inside the stator and are alternately arranged in a ring around the armature in sequence. The armature is fixed to the central moving platform of the flexible mechanism.

[0009] Preferably, there are two flexible constraint mechanisms in total. The driving main body mechanism is arranged 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 driving main body mechanisms are aligned with each other and fixed by first fixing screws; there are a plurality of connecting bumps on the outer circumferential surface of the stator; the connecting bumps are arranged in the grooves on the opposite side surfaces of the two cover plates and fixed by pre-tightening screws.

[0011] Preferably, the interior of the stator is provided with four long arms and four short arms that are alternately arranged in sequence around the center of the stator. Four exciting coils are respectively wound around the four long arms; four permanent magnets are respectively fixed on the end faces of the four short arms.

[0012] Preferably, the electromagnetic drive biaxial nano-positioning platform further includes a displacement detection component. The displacement detection component includes a plurality of capacitive displacement sensors fixed on the cover plate. The plurality of capacitive displacement sensors are respectively used to detect the displacements of the center moving platform in different directions.

[0013] Preferably, the flexible connection chain includes a plurality of flexible units connected in sequence.

[0014] In a second aspect, the present invention provides a design method for a biaxial nano-positioning platform, which is used to design the foregoing electromagnetic drive biaxial nano-positioning platform; the design method is as follows:

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

[0016] Step 2: Preset the number of optimal chains, and use the flexibility matrix method to constrain the overall displacement of the synthesized double-layer parallel flexible constraint mechanism. For a flexible connection chain, the end connected to the cover plate is defined as a fixed boundary constraint, and an external force is applied to the end connected to the central moving platform for topological optimization of the flexible mechanism.

[0017] The objective function of the topological optimization of the flexible mechanism is:

[0018]

[0019] s.t.: KU = F

[0020]

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

[0022] Among them, f obj represents the objective function of the optimization problem; x i is the design variable, representing the density of the i-th unit of the flexible connection chain grid; is the eigenvalue; U is the global displacement matrix of the flexible connection chain; K and M respectively represent the global stiffness matrix and mass matrix of the flexible connection chain; F represents the external force; Φ m is the m-th structural mode; V i represents the material volume of the i-th unit of the grid; V * is the total required material volume; u in,x , u in,y respectively represent the displacement components in the x and y axis directions; represents the input displacement constraint condition; ε * is the constraint index of the input end displacement; x min is the design variable of the empty body unit; n is the number of units; N dof is the order of the natural frequency.

[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 is the flexibility of one set of opposite flexible connection chains (11); T y (·), T z (·) are respectively the flexibility conversion matrices for rotation about the y-axis and z-axis.

[0026] The expression of the flexibility C p 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 is the flexibility matrix of the flexible connection branch chain (11).

[0029] Preferably, the driving main body mechanism satisfies the following conditions:

[0030]

[0031] Among them, N is the number of turns of the excitation coil; I m and |d m | are the maximum current and displacement respectively. u 0 is the vacuum permeability; d 0 represents the initial gap between the armature and the excitation coil; is the bias DC magnetic flux generated by the permanent magnet. B sat is the saturation magnetic flux density.

[0032] Preferably, the sensitivity of the objective function and all its constraints is processed by a gradient optimization algorithm.

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

[0034] 1. The present invention adopts Maxwell force electromagnetic drive, which has unique advantages such as non-contact, high force density and adjustable stroke, making it outstanding in multi-axis drive of hundreds of micrometers and providing a new solution for the drive of high-performance biaxial micro-nano positioning platform. Through the two-way progressive structural topology optimization method, the integrated design of configuration and parameters is effectively solved, further balancing the stroke and working bandwidth, and simplifying the complexity of multi-performance design calculation.

[0035] 2. The present invention uses Maxwell force electromagnetic drive, and utilizes the difference in electromagnetic attraction caused by the difference in magnetic field intensity 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 here, making the present invention have a wider driving ability. The same structural form can realize small-stroke, high-bandwidth drive similar to piezoelectric actuators, and can also independently realize millimeter-level large-stroke, low-bandwidth drive similar to voice coil motors.

[0036] 3. The present invention designs the nano-positioning platform through the flexible structure topology optimization design method, overcomes the difficulties of traditional multi-degree-of-freedom structure design, and improves the structural performance from the perspective of conceptual design. Its advantages are novel structure and good flexibility, which can accurately control micro-displacement movement and flexibly adjust the trajectory of the moving platform in space. At the same time, the double-layer flexible parallel mechanism design effectively solves the twisting phenomenon of the single-layer flexible structure during the driving process, improving the positioning accuracy. The double-layer flexible structure is arranged in a cross shape, allowing independent driving in the same direction, eliminating coupling, enhancing the control accuracy, and the movement trajectory can be realized by adjusting the parameters of the Maxwell force electromagnetic drive signal.

[0037] 4. The present invention adopts a typical PID controller with a feed-forward compensator, significantly improving the tracking performance of the system. By calculating the system error and adjusting the electromagnetic drive voltage signal of the Maxwell force, the drive system can effectively compensate for the motion error caused by external disturbances. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] To more clearly interpret the present invention patent, the drawings of the embodiments will be briefly introduced below. Obviously, these drawings only represent specific embodiments of the present invention. For those skilled in the art, other related drawings can be derived based on these drawings without additional creative work.

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

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

[0041] Figure 3 It is a schematic overall structure diagram of the dual-axis nano-positioning platform provided in Embodiment 1 of the present invention.

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

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

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

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

[0046] Figure 8 It is a schematic internal structure diagram of the dual-axis nano-positioning platform provided in Embodiment 1 of the present invention.

[0047] Figure 9 It is a system control block diagram of the dual-axis nano-positioning platform provided in Embodiment 2 of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[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 drawings.

[0049] Embodiment 1

[0050] See Figure 3As shown in the figure, a biaxial nano-positioning platform is divided into upper, middle, and lower layers, including two flexible constraint mechanisms and a driving main body mechanism arranged between the two flexible constraint mechanisms. The two flexible constraint mechanisms are respectively called the first flexible constraint mechanism and the second flexible constraint mechanism.

[0051] See Figure 6 As shown in the figure, the first flexible constraint mechanism serves as the upper layer of the biaxial nano-positioning platform and includes a first cover plate 2 and a flexible mechanism 12 installed inside the first cover plate 2.

[0052] See Figure 4 As shown in the figure, the driving main body mechanism serves as the middle layer of the biaxial nano-positioning platform and includes a stator 8, a permanent magnet 10, an exciting coil 13, and an armature 14.

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

[0054] A displacement detection component is also installed on the second cover plate 7. The displacement detection component includes a capacitive displacement sensor 1, a capacitive displacement sensor 6, and a capacitive displacement sensor 9; fixed through holes for installing the capacitive displacement sensor 1, the capacitive displacement sensor 6, and the capacitive displacement sensor 9 are provided on the second cover plate 7. The three capacitive displacement sensors are respectively used to detect the x and y direction displacements and the z-axis rotation angle of the central moving platform of the flexible mechanism 12.

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

[0056] In the first cover plate 2 and the second cover plate 7, the connection structures of the flexible mechanism 12 are the same; among them, the flexible mechanism 12 includes a central moving platform and four flexible connection chains 11 distributed around 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 surfaces of the first cover plate 2 or the second cover plate 7 through the flexible connection chains 11. See Figure 5 As shown in the figure, the flexible connection chains 11 are distributed in a cross-shaped pattern on the flexible mechanism 12.

[0057] The specific structure of the flexible connection chain 11 is obtained by solving through the biaxial nano-positioning platform design method in Embodiment 2.

[0058] See Figure 6 and 7 As shown in the figure, the two flexible mechanisms 12 are respectively located at the middle positions inside the first cover plate 2 and the second cover plate 7 and are symmetrically distributed up and down.

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

[0060] See Figure 6 and 7 As shown, the four excitation coils 13 are arranged in an orthogonal distribution; specifically, the four excitation coils 13 are evenly distributed circumferentially 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; corresponding permanent magnets 10 are interspersed between any two adjacent excitation coils 13; the permanent magnets 10 and the stator are bonded by structural adhesive.

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

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

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

[0064] The novel linear variable reluctance drive biaxial nano-positioning platform provided by this embodiment can achieve high bandwidth and large working space. Generally speaking, the main contributions include: in terms of precise drive, a biaxial nano-positioning platform based on Maxwell force electromagnetic drive is proposed. At the same time, the non-contact drive at the input end effectively avoids the cross-coupling phenomenon and can effectively protect the actuator. In addition, the nano-positioning platform provided by this embodiment adopts a double-layer flexible parallel mechanism design, which effectively solves the distortion phenomenon of a single flexible structure during the driving process, thereby improving the positioning accuracy of the nano-positioning platform.

[0065] Embodiment 2

[0066] A design method for a biaxial nano-positioning platform includes the following steps:

[0067] Problem description of the Maxwell force electromagnetic drive: As Figure 1As shown in part (a), when the exciting coil 13 is energized in the horizontal direction, the permanent magnet 10 and the windings of the exciting coil 13 will respectively generate a bias DC magnetic flux in the circuit and an 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 the exciting force generated between the left and right poles of the armature 14 ultimately generates a total exciting force F x . The detailed derivation of the total exciting force is as follows:

[0068]

[0069] Where Since in the working gap, the magnetic resistance of the permanent magnet 10 is significantly greater than that of air, therefore, the total bias magnetic flux generated by the movement of the armature 14 can be regarded as a constant. Here, represents the residual magnetic flux of the permanent magnet 10, where and A pm are respectively the magnetic flux and the pole face area of the permanent magnet 10. d 0 and d respectively represent the initial working air gap on each side and the displacement of the armature 14. As Figure 1 (b) shows, α is the magnetic leakage coefficient, and its expression is:

[0070]

[0071] Where R pm = L pm / (u 0 A pm ) and R L are respectively the internal magnetic resistance and the leakage magnetic resistance of the permanent magnet 10. The result of R L can be obtained through finite element analysis. The magnetic resistances of the two working air gaps between the armature 14 and the permanent magnet 10 are respectively R 1 =(d 0 - d) / (u 0 A) and R 2 =(d 0 + d) / (u 0 A). Here, u 0 is the vacuum permeability, and A represents the pole face area of the stator 8 iron core. In addition, R r = d 1 / (u 0 A) is the magnetic resistance of the air gap between the permanent magnet 10 and the armature 14, where d 1 is the length of the air gap between the permanent magnet 10 and the armature 14. u 0 is modeled as the magnetic resistance R r .

[0072] When the excitation coil 13 works independently, the total excitation coil NI generates an alternating magnetic flux in the circuit Alternating magnetic flux can be expressed as Therefore, based on the Maxwell stress tensor theory, the DC magnetic flux and the AC magnetic flux will be superimposed. According to the magnetic flux B in the right air gap 1 and the magnetic flux B in the left air gap 2 , the excitation force F can be obtained a as follows:

[0073]

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

[0075]

[0076] where, I m and |d m | are the maximum current and displacement respectively.

[0077] Description of the flexible mechanism topology optimization problem: As shown in part (a) of Figure 2 , for the synthesis process of the optimal flexible connection branch chain 11, it is mainly divided into two steps. In step I, the size and volume of the optimal branch chain are defined as constraints, and the objective function is set to maximize the natural frequency. Γ is the fixed boundary constraint. In step II, the number of the optimal branch chains is predetermined, and the overall displacement of the synthesized double-layer flexible parallel mechanism is constrained by the compliance matrix method. In step II, Γ is redefined as the fixed boundary constraint at the left end, and external forces F x and F y in the x-axis and y-axis directions of the single-layer flexible connection branch chain 11 are applied at the right end. To achieve the best structural optimization result of the single-layer flexible connection branch chain 11, the two-way progressive structural optimization method is adopted in this embodiment for the flexible mechanism design of the related nano-positioning platform, aiming to maximize the fundamental frequency.

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

[0079] ① The objective function of the flexible connection branch chain 11: Maximize the natural frequency.

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

[0081] ③ The expected working space of the flexible connection branch chain 11: 240×240μm 2 。

[0082] ④ The size of the entire mechanism: 125×35mm 2 。

[0083] Structural arrangement: In order to convert the local displacement u j constraints into global displacements Δ x and Δ y constraints, the flexible connection branch chain 11 can be revised into a flexibility matrix C. According to Hooke's law, the elastic deformation control equation of the flexible connection branch chain 11 can be expressed as:

[0084]

[0085] where δ and F represent the displacement vector and the load vector respectively. θ z is the global rotation angle about the z-axis; are the translational flexibilities along the x, y, and z axes and the rotational flexibility about the z-axis direction respectively.

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

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

[0088] where, T Z (0) represents the flexibility transformation matrix from the local system of the flexible connection branch chain 11 to the O xyz system with an angle of 0°; C l =T Z (0)CT Z (0) T represents the flexibility matrix of the flexible connection branch chain 11 in O xyz ; T y (π) represents the flexibility transformation matrix for a rotation operation with an angle of π about the O y axis.

[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 T y (π / 2) T -1 -1 (7)

[0091] where T z (π / 2) represents the flexibility transformation matrix for a rotation operation of angle π / 2 about the O z axis.

[0092] Since the double-layer flexible mechanism 12 of the biaxial nanopositioning platform is in a parallel configuration, its flexibility in the O xyz system can be defined as:

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

[0094] where K in is the input stiffness of the nanopositioning platform.

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

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

[0097] where the driving force F in,x =F in,y =F a .

[0098] Performance modeling: In this optimization, the volume constraint applies to the flexible connection link 11, while the displacement constraint involves the biaxial nanopositioning platform. The objective function focuses on maximizing the natural frequency of the flexible connection link 11 rather than that 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 introduced in detail in the following description. ​​

[0099] The above optimization problem requires solving an eigenvalue problem. The governing equation 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; and u(t) represent the vectors of acceleration, velocity, and transient displacement, respectively; v 0 and u 0 are the initial velocity and initial displacement, respectively. When the damping matrix C = 0 and the external force F(t) = 0, this equation is the undamped free vibration equation. By solving this equation, the natural frequencies and mode shapes of the final eigenvalue problem can be defined as:

[0102]

[0103] where m is the set of modes corresponding to the degrees of freedom of the structure, and Φ m is the m-th eigenvalue and the eigenvector.

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

[0105]

[0106] where E(x i ) is the interpolated Young's modulus; ρ(x i ) is the design variable corresponding to the interpolated density; p = 3 is the penalty factor used to prevent the occurrence of intermediate densities; E 0 and ρ 0 represent the Young's modulus and density of the solid material, respectively. The design variable x i represents the density of the i-th element, and a small value x min .

[0107] Through the interpolation method, the global stiffness matrix K and the global mass matrix M can be defined as:

[0108]

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

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

[0111]

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

[0113]

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

[0115]

[0116] ε * is a very small positive number. Therefore, if it means the displacement constraint is satisfied; otherwise, it means the displacement constraint is not satisfied.

[0117] Optimization model: The flexible mechanism topology optimization problem for constructing the flexible connection branch chain 11 with the optimal performance can be formulated as:

[0118]

[0119] where, f obj represents the objective function of the optimization problem. The design variable x i represents the density of the i-th grid cell in the design domain of the flexible connection branch chain 11. A small value x min (for example, 0.001 instead of 0 to represent a hollow cell) is used as the design variable of the optimization problem. λ j = ω j 2 where ω j 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 * represent the total material volume and the required material volume respectively. u in and represent the magnitude of the input displacement and its constraint conditions respectively. n and N dof represent the number of elements and the order of the natural frequency respectively.

[0120] Sensitivity analysis: The dynamic topology optimization problem with multiple constraints is usually processed by the gradient optimization algorithm to calculate the sensitivities of the objective function and all its constraints. Therefore, the sensitivity of the objective function f obj can be expressed by Equation (18), and its form is as follows:

[0121]

[0122] By substituting the eigenvector Φ mNormalize with respect to the mass matrix M. Thus, the sensitivity of the i-th natural frequency in the solid-void design can be calculated as:

[0123]

[0124] In the two-way progressive structural optimization method, only two discrete design variables are used, namely solid element 1 and void element x min . At the same time, the sensitivity values used represent the element sensitivity. Therefore, the sensitivity values of the solid-void elements can be expressed as:

[0125]

[0126] When x min approaches 0, the sensitivity values of the solid-void elements of the objective function can be simplified to:

[0127]

[0128] The change in displacement can be estimated based on the change in the design variable. Therefore, for the displacement constraint, the sensitivity analysis can be expressed as:

[0129]

[0130] where U i is the displacement vector caused by the i-th element, which is obtained by U = K -1 F i ; U ij 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 i+1 in the next iteration can be approximately estimated by the displacement u in i in the current iteration, expressed as:

[0132]

[0133] where u in i and u in i+1 are the displacements of the current iteration and the next iteration respectively. For specific descriptions, relevant literature can be referred to.

[0134] When x min approaches 0, the sensitivity values of the solid-void elements of the displacement constraint can be simplified to:

[0135]

[0136] The above is the design of the biaxial nano-positioning platform driver and the optimization model of the flexible mechanism. Through the above process, the biaxial nano-positioning platform of the present invention can be effectively designed, as Figure 3 shown.

[0137] In addition, as shown in Figure 9 the present invention applies a typical PID controller with a feedforward compensator to the biaxial nano-positioning platform, effectively improving the performance of the overall system for precise motion tracking, positioning, etc.

[0138] According to Figure 1 , when currents are applied to the left and right coils, the generated Maxwell electromagnetic force will drive the end effector to move left and right; while when currents are applied to the upper and lower coils, the end effector can be made to move up and down.

[0139] The above description is only a schematic elaboration of the present invention, which can be understood and implemented by ordinary technicians in the field without creative work. Through this description, technicians can understand various implementation manners and implement the present invention with the aid of necessary software and hardware. Based on this understanding, the contribution of the technical solution can be embodied in the form of a software product, and the software can be stored in a computer-readable storage medium.

[0140] The above is only the best embodiment of the present invention and does not limit the scope of the present invention. Any modification, equivalent replacement or improvement made within the spirit and principle of the present invention shall be included within the protection scope of the present invention. The protection scope shall be determined according to the content described in the claims.

[0141] In summary, the present invention provides a Maxwell force electromagnetic-driven biaxial nano-positioning 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. An electromagnetically driven dual-axis nanopositioning platform, comprising a flexible constraint mechanism and a driving main body mechanism; characterized in that: The flexible constraint mechanism comprises a cover plate and a flexible mechanism (12) installed in the cover plate; the flexible mechanism (12) comprises a central mobile platform and four flexible connecting branches (11) distributed around the central mobile platform; two ends of the flexible connecting branches (11) are respectively fixed to the central mobile platform and the cover plate; The driving main body mechanism comprises a stator (8), a permanent magnet (10), an excitation coil (13) and an armature (14); the armature (14) is located at the center of the stator (8); four permanent magnets (10) and four excitation coils (13) are fixed in the stator (8) and are alternately arranged in sequence around the armature (14) to form a ring; the armature (14) is fixed to the central moving platform of the flexible mechanism (12).

2. The electromagnetically driven dual-axis nanopositioning platform according to claim 1, characterized in that: There are two flexible constraint mechanisms in total; the driving main body mechanism is arranged between the two flexible constraint mechanisms; and the flexible mechanisms (12) in the two flexible constraint mechanisms are aligned with each other.

3. The electromagnetically driven dual-axis nanopositioning platform according to claim 2, characterized in that: The cover plates in the two driving main body mechanisms are aligned with each other and fixed by a first fixing screw (3); a plurality of connecting protrusions are provided on the outer circumferential surface of the stator (8); the connecting protrusions are arranged in grooves on opposite sides of the two cover plates (7) and fixed by a pre-tightening screw (4).

4. The electromagnetically driven dual-axis nanopositioning platform according to claim 1, characterized in that: The stator (8) is provided with four long arms and four short arms arranged alternately in sequence around the center of the stator (8); four excitation coils (13) are respectively wound on the four long arms; and four permanent magnets (10) are respectively fixed on the end surfaces of the four short arms.

5. The electromagnetically driven dual-axis nanopositioning platform according to claim 1, characterized in that: It also includes a displacement detection component; the displacement detection component includes a plurality of capacitive displacement sensors fixed on the cover plate; the plurality of capacitive displacement sensors are respectively used to detect the displacement of the central moving platform in different directions.

6. The electromagnetically driven dual-axis nanopositioning platform according to claim 1, characterized in that: The flexible connecting branch (11) comprises a plurality of flexible units connected in sequence.

7. A method for designing a dual-axis nanopositioning platform, characterized in that: Used to design an electromagnetically driven dual-axis nanopositioning platform as claimed in claim 1; the design method is as follows: Step 1: Define the size and volume of the optimal flexible connection branch as constraints, and set the objective function to maximize the natural frequency; Step 2: predetermine the number of optimal branches, and constrain the overall displacement of the synthesized double-layer parallel flexible constraint mechanism through the flexibility matrix method; for a flexible connection branch, define a fixed boundary constraint at the end of the connecting cover plate, and apply external force to the end connected to the central mobile platform to perform topological optimization of the flexible mechanism; The objective function of topology optimization of flexible mechanisms is: Among them, f obj represents the objective function of the optimization problem; x i is the design variable; is the eigenvalue; U, K and M represent the global displacement matrix, stiffness matrix and mass matrix respectively; F represents the external force; Φ m is the structural mode; V i Represents the material volume of the grid cell; V * is the total material volume required; u in,x 、u in,y Represent the displacement components in the x-axis and y-axis directions respectively; represents the input displacement constraint; ε * is the constraint index of the input displacement; x min is the design variable of the hollow unit; n is the number of units; N dof is the natural frequency order.

8. A method for designing a dual-axis nanopositioning platform according to claim 7, characterized in that: The flexibility expression of the flexible mechanism (12) is: C s =[(C p ) -1 +[T z (π / 2)C p T y (π / 2) T ] -1 ] -1 Among them, C p is the flexibility of one set of relatively flexible connecting branches (11); T y (·),T z (·) are the flexibility transformation matrices for rotation around the y-axis and z-axis respectively; Flexibility C p The expression is as follows: C p =[(C l ) -1 +[T y (π)C l T y (π) T ] -1 ] -1 Among them, C l is the flexibility matrix of the flexible connecting branch (11).

9. A dual-axis nanopositioning platform design method according to claim 7, characterized in that: The driving main body mechanism meets the following conditions: Wherein, N is the number of turns of the excitation coil (13); I m and |d m | are the maximum current and displacement respectively; u0 is the vacuum magnetic permeability; d0 represents the initial gap between the armature (14) and the excitation coil (13); B is the bias DC magnetic flux generated by the permanent magnet (10); sat is the saturation flux density.

10. The method for designing a dual-axis nanopositioning platform according to claim 7, characterized in that: The sensitivity of the objective function and all its constraints is handled by a gradient optimization algorithm.

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