Dual-axis flexible guide mechanism for nanopositioning platform and its stiffness modeling method

Abstract

The present invention discloses a dual-axis flexible guide mechanism for a nano-positioning platform and a stiffness modeling method thereof. The dual-axis flexible guide mechanism comprises a workbench base and a fixed frame, wherein the workbench base is set in the middle of the fixed frame, and 8 symmetrically distributed sinusoidal flexible hinges of the same structure are arranged in the middle and connected to the fixed frame. The stiffness modeling method comprises the following steps: using the second theorem of Cardinality to derive the flexibility of the sinusoidal flexible hinge unit, and then further using a matrix-based flexibility modeling method to obtain the stiffness matrix of the flexible hinge, and finally using the same method to obtain the total stiffness of the mechanism. The present invention utilizes the characteristics of the sinusoidal flexible hinge that can undergo elastic elongation and bending deformation at the same time, and develops a dual-axis flexible guide mechanism for a nano-positioning platform, which can realize the decoupled motion of the two axes and has a large motion stroke, thereby reducing the overall size and motion quality of the platform and improving the working bandwidth of the positioning platform.

Description

Technical Field

[0001] The invention belongs to the technical field of precision manufacturing, and in particular relates to a dual-axis flexible guiding mechanism for a nano-positioning platform and a stiffness modeling method thereof. Background Art

[0002] The nanopositioning platform is a motion, positioning and manipulation platform with nanometer-level precision. It is widely used in nano-measurement, nano-processing, bio-manufacturing and other fields. It is one of the core modules of nano-manufacturing equipment. Its development level directly affects the development level of technology in the field of micro-science and has extremely important practical research significance. The guide mechanism is the main component of the nanopositioning platform. The compactness, decoupling, motion resolution, and mass of the moving parts of the structure directly determine the performance of the nanopositioning platform. The most commonly used structure in the guide mechanism is the flexible hinge, which uses the elastic deformation of the material to achieve structural deformation and output displacement. Because of its advantages of no mechanical friction, no gap, high motion sensitivity, compact structure, small size, and light weight, it is widely used in various types of nanopositioning platforms.

[0003] For flexible guide mechanisms with dual degrees of freedom, the current mainstream design method is to combine flexible hinges with unidirectional degrees of freedom in series and parallel. The flexible hinges with unidirectional degrees of freedom that have been developed include right-angle profile type, elliptical profile type, transitional rounded profile type, and circular profile type. However, the flexible guide mechanism formed by combining such flexible hinges will inevitably increase the structural size and motion mass of the platform. In order to improve working performance, another method is to directly use flexible hinges with dual degrees of freedom in parallel, but the existing flexible hinges with dual degrees of freedom, such as arc type and corrugated spring type, all face the common problem that the elastic deformation that can be produced by the hinge per unit length is limited. Therefore, how to continuously break through this limit is one of the key technologies in the study of flexible guide mechanisms. Summary of the invention

[0004] The object of the present invention is to provide a dual-axis flexible guiding mechanism for a nanopositioning platform and a stiffness modeling method thereof.

[0005] Technical solution to achieve the purpose of the present invention:

[0006] A dual-axis flexible guide mechanism for a nanopositioning platform, comprising a fixed frame, a plurality of flexible hinges, and a workbench base.

[0007] The workbench base is located in the middle of the fixed frame, and at least one of the flexible hinges is respectively connected between the four outer side surfaces of the workbench base and the inner side surface of the fixed frame, and the flexible hinge is a sinusoidal flexible hinge.

[0008] Furthermore, it also includes a driving module, a supporting structure and a workbench, the bottom of the fixed frame is connected to the supporting structure, and the top of the workbench base is connected to the workbench.

[0009] Furthermore, the fixed frame includes a plurality of frame mounting holes, the workbench base includes a plurality of workbench mounting holes, the bottom of the fixed frame is connected to the supporting structure by bolts, and the top of the workbench base is connected to the workbench by bolts.

[0010] Furthermore, it also includes an armature, the bottom surface of the workbench base is the armature mounting surface, and the armature is connected to the armature mounting surface by gluing.

[0011] Furthermore, it also includes a sensor mounting seat, which is installed on the top of the workbench base and has a sensor mounting hole.

[0012] Furthermore, the free end of the flexible hinge is connected to the workbench base, and the fixed end of the flexible hinge is connected to the fixed frame.

[0013] Furthermore, the number of the flexible hinges is eight, and the eight flexible hinges are divided into four groups and symmetrically arranged on four outer side surfaces of the workbench base, and each group of flexible hinges is vertically arranged with respect to the flexible hinges of the adjacent group.

[0014] According to the above-mentioned stiffness modeling method of the dual-axis flexible guide mechanism for the nanopositioning platform, the following steps are included:

[0015] Step 1: Determine the functional expression of the sine curve formed by the center line of a single sinusoidal flexible hinge:

[0016] A rectangular coordinate system o0-x0y0z0 is established at the center point of the fixed end of the sine flexible hinge, where A, p x and n are used to represent the amplitude of the sine curve, the length of a cycle, and the number of cycles of a sine flexible hinge, respectively. The function expression of the sine curve formed by the center line of the sine flexible hinge is:

[0017]

[0018] Step 2: Determine the normal stress and shear stress at any position of a single period sine curve:

[0019] Choose a periodic sinusoidal flexible hinge x∈[0,p x ], its free end load is F = [f x ,f y ,f z ,m x ,m y ,m z ]T , where f x ,f y ,f z Respectively represent the forces on the free end of the flexible hinge in the x0, y0, and z0 directions for one period, m x ,m y ,m z They represent the torques on the free end of the flexible hinge in the x0, y0, and z0 directions in a period, and the deformation of the free end is u=[u x ,u y ,u z ,θ x ,θ y ,θ z ] T , where u x ,u y ,u z Respectively represent the free end displacement of the free end of the flexible hinge in the x0, y0, and z0 directions for one period, θ x ,θ y ,θ z Represent the rotation angles of the free end of a periodic flexible hinge in the x0, y0, and z0 axes, respectively. The bending torque M at any x0 position is b for:

[0020] M b =[M xy ,M xz ,M yz ]

[0021] in,

[0022]

[0023] Among them, M xy , M xz , M yz They represent the bending torques in the x0y0, x0z0, and y0z0 planes at any x0 position respectively. The expressions of the plane normal force N and shear force S at any x0 position are as follows:

[0024]

[0025] in,

[0026]

[0027] Step 3: Determine the elastic strain energy of the flexible hinge for a single cycle:

[0028] According to Cartesian's second theorem, the elastic strain energy U generated by deformation is expressed as

[0029]

[0030] in is the second moment of inertia of the sinusoidal flexure hinge cross section, A c =wt is the cross-sectional area of ​​the sinusoidal flexible hinge, where w represents the width of the flexible hinge, t represents the thickness of the flexible hinge, and E represents the Young's modulus of the material. is the corresponding shear modulus, μ represents the Poisson's ratio of the material, α t represents the shear coefficient of the rectangular cross section, L is the actual length of a single period flexure hinge along the sinusoidal direction;

[0031] Step 4: Determine the deformation of the flexible hinge in a single cycle:

[0032] According to the second theorem of Castelluccio, the spatial flexibility expression of the sinusoidal flexible hinge is as follows:

[0033]

[0034] in

[0035]

[0036] Therefore, the relationship between the deformation of a single cycle flexible hinge and the end load is: u = C s F;

[0037] Step 5: Determine the spatial flexibility matrix of a single sinusoidal flexible hinge:

[0038] A single sinusoidal flexible hinge is composed of n periodic flexible hinge units connected in series. The spatial flexibility matrix of the first random sinusoidal flexible hinge is obtained according to the spatial position transformation method of the flexibility matrix:

[0039]

[0040] Where T i From the coordinate system o i -x i y i z i to c1 -x c1 y c1 z 0c1 The position transformation matrix, o c1 -x c1 y c1 z 0c1 is the coordinate system of the free end of the flexible hinge, point o c1 Represents the global coordinate origin of a single sinusoidal flexure hinge, located at the free end, point o i represents the local coordinate origin of the i-th period flexible hinge, T i Expressed as

[0041]

[0042] Where R i (α i ) is from the coordinate system o i -x i y i z i to c1 -x c1 y c1 z 0c1 The rotation matrix, S i (r i ) is r i The skew symmetry operator of is α, whose rotation angle around the z axis is i , R i (α i ) is defined as

[0043]

[0044] r i =[x i ,y i ,z i ] is in the coordinate system o c1 -x c1 y c1 z 0c1 The point o represented in i The location of S i (r i ) is given by

[0045]

[0046] Since the single-period flexible hinges are in series in a single sinusoidal flexible hinge, the position of the i-th periodic flexible hinge is in the coordinate system o c1 -x c1 y c1 z 0c1 Indicated as r i =[-ip x ,0,0], and the rotation angle around the z-axis is α i =0;

[0047] Step 6: Determine the stiffness of the entire dual-axis flexible guide mechanism:

[0048] The stiffness at the center of the mechanism is expressed as

[0049]

[0050] Among them, T j is the angle around the z-axis The position transformation matrix, that is, the free end coordinate system of the flexible hinge involved in the calculation o cj -x cj y cj z cj To o-xyz position conversion matrix, j represents the number of flexible hinges, j = 1, 2, 3, 4, o-xyz is the coordinate system at the center of the workbench base,

[0051] When modeling the flexibility in the x and y directions, the workbench base is regarded as a rigid body. Considering the symmetric characteristics of the dual-axis flexible guide mechanism about the x and y axes, the stiffness in the x and y directions at the center of the mechanism is expressed as k x =k y =K(1,1)=K(2,2).

[0052] Compared with the prior art, the present invention has the following significant advantages:

[0053] The present invention designs a sinusoidal flexible hinge, which increases the motion stroke and reduces the motion mass under the same size. At the same time, this hinge has good dual-axis motion guiding ability. Using this hinge to construct a two-axis nanopositioning platform can give full play to the working performance of the flexible hinge, greatly reduce the structural size and motion mass of the platform, and improve the working performance of the platform. At the same time, a corresponding stiffness analytical model is established, which can provide guidance in the design stage, simplify the design process, and improve efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a partial structural schematic diagram of the dual-axis flexible guiding mechanism for the nano-positioning platform of the present invention.

[0055] Figure 2 It is a rear view of the dual-axis flexible guide mechanism for the nanopositioning platform of the present invention.

[0056] Figure 3 It is a front view of a sinusoidal flexible hinge of a dual-axis flexible guide mechanism for a nanopositioning platform of the present invention.

[0057] Figure 4 It is a structural schematic diagram of a sinusoidal flexible hinge of a dual-axis flexible guide mechanism for a nanopositioning platform of the present invention.

[0058] Figure 5 It is a structural schematic diagram of a dual-axis flexible guiding mechanism for a nano-positioning platform of the present invention.

[0059] In the figure, 1. fixed frame, 2. flexible hinge, 3. workbench base, 4. frame mounting hole, 5. workbench mounting hole, 6. armature mounting surface, 7. drive module, 8. support structure, 9. workbench, 10. sensor mounting seat, 11. sensor mounting hole. DETAILED DESCRIPTION

[0060] The present invention is further described in detail below in conjunction with the accompanying drawings.

[0061] The structure of the dual-axis flexible guide mechanism for the nanopositioning platform of the present invention is as follows: Figure 1 , 5 As shown, it includes a fixed frame 1, a workbench base 3 in the middle of the frame 1, and the two are connected by 8 identical sinusoidal flexible hinges that are symmetrically distributed. The fixed frame 1 has a frame mounting hole 4 for mounting and positioning. The workbench base 3 has a workbench mounting hole 5 for connecting to a workbench 9.

[0062] The working principle of the present invention is: Figure 5 First, the fixed frame 1 is connected to the corresponding support structure 8 by bolting through the frame mounting holes 4, and then the workbench 9 is installed on the workbench base 3 by bolting through the workbench mounting holes 5, and then the armature is connected to the armature mounting surface 6 by gluing, and finally the sensor mounting seat 10 is installed on the fixed frame 1 by bolting, and the sensor mounting seat 10 has a useful sensor mounting hole 11. In this way, when the driving module 7 (specifically the normal force electromagnetic driving module, which is used to generate the Maxwell electromagnetic force to act on the armature) acts on the armature, the armature will drive the workbench base 3 to move together, that is, the free end of the sinusoidal flexible hinge will be affected by the force of the driving module 7 and produce elastic deformation. Different from the traditional two-axis mechanism, the free end and fixed end of the sinusoidal flexible hinge are connected to the workbench base 3 and the fixed frame 1 respectively. When driven, axial extension and / or compression and plane bending will occur simultaneously. Using 8 such identical flexible hinges 2 to form two groups of mutually perpendicular parallelogram structures can achieve good two-axis decoupled motion and suppress unwanted additional motion. In this way, the working performance of the sinusoidal flexible hinge can be fully utilized, avoiding the problems of excessive size structure and motion mass caused by the single degree of freedom of ordinary flexible hinges in traditional flexible mechanisms.

[0063] Rigidity is an important parameter of the flexible guide mechanism. When the driving force is constant, its magnitude directly determines the working stroke of the positioning platform. In order to provide a reference model for the optimal design of the flexible guide mechanism proposed in the present invention, the present invention establishes the flexibility matrix of the sinusoidal flexible hinge.

[0064] The steps of the flexible mechanism stiffness modeling method are as follows:

[0065] Step 1: Determine the functional expression of the sine curve formed by the center line of a single sinusoidal flexible hinge 2:

[0066] according to Figure 3The rectangular coordinate system o0-x0y0z0 is established at the center point of the left fixed end of the sinusoidal flexible hinge in the manner shown in the figure, where A, p x and n are used to represent the amplitude of the sine curve, the length of a cycle, and the number of cycles of a sine hinge, respectively. Then the function expression of the sine curve formed by the center line of the sine flexible hinge is

[0067]

[0068] Step 2: Determine the normal stress and shear stress at any position in a single cycle:

[0069] Choose a periodic sinusoidal flexible hinge x∈[0,p x ], its free end load is F = [f x ,f y ,f z ,m x ,m y ,m z ] T , where f x ,f y ,f z Respectively represent the forces on the free end of the flexible hinge in the x0, y0, and z0 directions for one period, m x ,m y ,m z They represent the torques on the free end of the flexible hinge in the x0, y0, and z0 directions in a period, and the deformation of the free end is u=[u x ,u y ,u z ,θ x ,θ y ,θ z ] T , where u x ,u y ,u z Respectively represent the free end displacement of the free end of the flexible hinge in the x0, y0, and z0 directions for one period, θ x ,θ y ,θ z Represent the rotation angles of the free end of a periodic flexible hinge in the x0, y0, and z0 axes, respectively. The bending torque M at any x0 position is b for:

[0070] M b =[M xy ,M xz ,M yz ]

[0071] in

[0072]

[0073] Among them, M xy , M xz , M yz They represent the bending torques in the x0y0, x0z0, and y0z0 planes at any x position, respectively. The expressions of the plane normal force N and shear force S at any x0 position are as follows:

[0074]

[0075] in

[0076]

[0077] Step 3: Determine the elastic strain energy of the flexible hinge for a single cycle:

[0078] According to Cartesian's second theorem, the elastic strain energy U generated by deformation is expressed as

[0079]

[0080] in is the second moment of inertia of the sinusoidal flexure hinge cross section, and A c =wt is the cross-sectional area of ​​the sinusoidal flexure hinge, where Figure 4 The w shown here represents the width of the flexible hinge, as Figure 3 The t shown here represents the thickness of the flexible hinge. In addition, E represents the Young's modulus of the material, is the corresponding shear modulus, μ represents the Poisson's ratio of the material, α t represents the shear coefficient of the rectangular cross section, and L is the actual length of a single period flexure hinge along the sinusoidal direction.

[0081] Step 4: Determine the deformation of the flexible hinge in a single cycle:

[0082] According to the second theorem of Castelluccio, the spatial flexibility expression of the sinusoidal flexible hinge is as follows:

[0083]

[0084] in

[0085]

[0086] Therefore, the relationship between the hinge deformation and the end load in a single cycle is: u = C s F

[0087] Step 5: Determine the spatial flexibility matrix of a single sinusoidal flexure hinge

[0088] A single sinusoidal flexible hinge can be regarded as n (here 3) periodic flexible hinge units connected in series. According to the spatial position transformation method of the flexibility matrix, the spatial flexibility matrix of the random first sinusoidal flexible hinge can be obtained:

[0089]

[0090] Where T i From the coordinate system o i -x i y i z i to c1 -x c1 y c1 z 0c1 The position transformation matrix, o c1 -x c1 y c1 z 0c1 is the coordinate system of the free end of the flexible hinge, point o c1 Represents the global coordinate origin of a single sinusoidal flexure hinge, located at the free end, point o i represents the local coordinate origin of the i-th flexible hinge, Figure 3 The geometric position relationship between the single periodic sinusoidal flexible hinge and the single sinusoidal flexible hinge shown is determined and expressed as

[0091]

[0092] Where R i (α i ) is from the coordinate system o i -x i y i z i to c1 -x c1 y c1 z 0c1 The rotation matrix, S i (r i ) is r i The skew symmetry operator of is α, whose rotation angle around the z axis is i , R i (α i ) is defined as

[0093]

[0094] r i =[x i ,y i ,z i ] is in the coordinate system o c1 -x c1 y c1 z 0c1 The point o represented ini The location of S i (r i ) is given by

[0095]

[0096] Since the flexible single-period flexible hinges are in series in a single sinusoidal flexible hinge, Figure 3 It can be concluded that the position of the flexible hinge in the i-th period is in the coordinate system o c1 -x c1 y c1 z 0c1 It can be expressed as r i =[-ip x ,0,0], and the rotation angle around the z-axis is α i =0.

[0097] Step 6: Determine the stiffness of the entire dual-axis flexible guide mechanism

[0098] The biaxial flexible guide mechanism is composed of eight parallel sinusoidal flexible hinges connected in parallel, and the flexible guide mechanism is symmetrical about the x and y axes. In this way, when calculating the stiffness of the center of the mechanism according to the spatial position transformation method of the flexibility matrix, only half of the hinges can be calculated, and the final stiffness is twice that of this case. Therefore, the stiffness at the center of the mechanism can be expressed as

[0099]

[0100] At this time, T j is the angle around the z-axis The position transformation matrix is Figure 2 The free end coordinate system of the flexible hinge involved in the calculation is shown in cj -x cj y cj z cj (j represents the number of flexible hinge groups, j = 1, 2, 3, 4) to the position transformation matrix of o-xyz (the coordinate system at the center of the workbench base).

[0101] Although the workbench base rigidly connected to the flexible hinge has a certain deformation during the movement, this has little effect on the stiffness of the planar motion. Therefore, it can be regarded as a rigid body when modeling the flexibility in the x and y directions. Finally, considering the symmetric characteristics of the dual-axis flexible guide mechanism about the x and y axes, the stiffness in the x and y directions at the center of the mechanism can be expressed as k x =k y =K(1,1)=K(2,2).

[0102] The above shows and describes the basic principles, main features and advantages of the present invention. It should be understood by those skilled in the art that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A stiffness modeling method for a dual-axis flexible guide mechanism for a nanopositioning platform, the dual-axis flexible guide mechanism comprising a fixed frame (1), a plurality of flexible hinges (2), and a workbench base (3). The workbench base (3) is located in the middle of the fixed frame (1), and at least one of the flexible hinges (2) is respectively connected between the four outer side surfaces of the workbench base (3) and the inner side surface of the fixed frame (1), and the flexible hinge (2) is a sinusoidal flexible hinge; It also comprises a driving module (7), a supporting structure (8) and a workbench (9), wherein the bottom of the fixed frame (1) is connected to the supporting structure (8), and the top of the workbench base (3) is connected to the workbench (9); The fixed frame (1) includes a plurality of frame mounting holes (4), the workbench base (3) includes a plurality of workbench mounting holes (5), the bottom of the fixed frame (1) is connected to the support structure (8) by bolts, and the top of the workbench base (3) is connected to the workbench (9) by bolts; It also includes an armature, the bottom surface of the workbench base (3) is an armature mounting surface (6), and the armature is connected to the armature mounting surface (6) by gluing; It also includes a sensor mounting seat (10), the sensor mounting seat (10) is mounted on the top of the workbench base (3), and a sensor mounting hole (11) is provided on the sensor mounting seat (10); The free end of the flexible hinge (2) is connected to the workbench base (3), and the fixed end of the flexible hinge (2) is connected to the fixed frame (1); The number of the flexible hinges (2) is eight, and the eight flexible hinges (2) are divided into four groups and symmetrically arranged on four outer sides of the workbench base (3), and each group of flexible hinges (2) is arranged perpendicular to the flexible hinges (2) of the adjacent group; It is characterized in that The method comprises the following steps: Step 1: Determine the functional expression of the sine curve formed by the center line of a single sinusoidal flexible hinge: A rectangular coordinate system o0-x0y0z0 is established at the center point of the fixed end of the sinusoidal flexible hinge, where A, p x and n are used to represent the amplitude of the sine curve, the length of a cycle, and the number of cycles of a sinusoidal flexible hinge, respectively. The function expression of the sine curve formed by the center line of the sinusoidal flexible hinge is: Step 2: Determine the normal stress and shear stress at any position of a single period sine curve: Choose a periodic sinusoidal flexible hinge x∈[0,p x ], its free end load is F = [f x ,f y ,f z ,m x ,m y ,m z ] T , where f x ,f y ,f z Respectively represent the forces on the free end of the flexible hinge in the x0, y0, and z0 directions for one period, m x ,m y ,m z They represent the torques on the free end of the flexible hinge in the x0, y0, and z0 directions in a period, and the deformation of the free end is u=[u x ,u y ,u z ,θ x ,θ y ,θ z ] T , where u x ,u y ,u z Respectively represent the free end displacement of the free end of the flexible hinge in the x0, y0, and z0 directions for one period, θ x ,θ y ,θ z Represent the rotation angles of the free end of a periodic flexible hinge in the x0, y0, and z0 axes, respectively. The bending torque M at any x0 position is b for: M b =[M xy ,M xz ,M yz ] in, Among them, M xy , M xz , M yz They represent the bending torques in the x0y0, x0z0, and y0z0 planes at any x0 position respectively. The expressions of the plane normal force N and shear force S at any x0 position are as follows: in, Step 3: Determine the elastic strain energy of the flexible hinge for a single cycle: According to Castigliano's second theorem, the elastic strain energy U generated by deformation is expressed as in is the second moment of inertia of the sinusoidal flexure hinge cross section, A c =wt is the cross-sectional area of ​​the sinusoidal flexible hinge, where w represents the width of the flexible hinge, t represents the thickness of the flexible hinge, and E represents the Young's modulus of the material. is the corresponding shear modulus, μ represents the Poisson's ratio of the material, α s represents the shear coefficient of the rectangular cross section, L is the actual length of a single period flexure hinge along the sinusoidal direction; Step 4: Determine the deformation of the flexible hinge in a single cycle: According to Castigliano's second theorem, the spatial flexibility expression of the sinusoidal flexible hinge is as follows in Therefore, the relationship between the deformation of a single cycle flexible hinge and the end load is: y = C s F; Step 5: Determine the spatial flexibility matrix of a single sinusoidal flexible hinge: A single sinusoidal flexible hinge is composed of n periodic flexible hinge units connected in series. The spatial flexibility matrix of the first random sinusoidal flexible hinge is obtained according to the spatial position transformation method of the flexibility matrix: Where T i From the coordinate system o i -x i y i z i to c1 -x c1 y c1 z c1 The position transformation matrix, o c1 -x c1 y c1 z c1 is the coordinate system of the free end of the flexible hinge, point o c1 Represents the global coordinate origin of a single sinusoidal flexure hinge, located at the free end, point o i represents the local coordinate origin of the i-th period flexible hinge, T i Expressed as Where R i (α i ) is from the coordinate system o i -x i y i z i to c1 -x c1 y c1 z c1 The rotation matrix, S i (r i ) is r i The skew symmetry operator of is α, whose rotation angle around the z axis is i , R i (α i ) is defined as r i =[x i ,y i ,z i ] is in the coordinate system o c1 -x c1 y c1 z c1 The point o represented in i The location of S i (r i ) is given by Since the single-period flexible hinges are in series in a single sinusoidal flexible hinge, the position of the i-th periodic flexible hinge is in the coordinate system o c1 -x c1 y c1 z c1 Indicated as r i =[-ip x ,0,0], and the rotation angle around the z-axis is α i =0; Step 6: Determine the stiffness of the entire dual-axis flexible guide mechanism: The stiffness at the center of the mechanism is expressed as Among them, T j is the angle around the z-axis The position transformation matrix, that is, the free end coordinate system of the flexible hinge involved in the calculation o cj -x cj y cj z cj To o-xyz position conversion matrix, j represents the number of flexible hinges, j = 1, 2, 3, 4, o-xyz is the coordinate system at the center of the workbench base, When modeling the flexibility in the x and y directions, the workbench base is regarded as a rigid body. Considering the symmetric characteristics of the dual-axis flexible guide mechanism about the x and y axes, the stiffness in the x and y directions at the center of the mechanism is expressed as k x =k y =K(1,1)=K(2,2).

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