A fin effect jaw force-deformation theoretical modeling method based on a co-rotation model

Abstract

The application discloses a fin effect clamp jaw force-deformation theoretical modeling method based on a co-rotation model, decomposes a fin effect clamp jaw structure into a plurality of beam units, decouples local deformation and rigid body motion of each beam unit through co-rotation theoretical modeling, respectively establishes an axial deformation model and a rotational deformation model, and introduces an effective length correction coefficient to consider the influence of physical hinge connection; a tangent stiffness matrix of the beam unit is constructed and assembled into a global tangent stiffness matrix; a Newton-Raphson iteration method is used to realize force balance solving of a load increment step, and a global force-displacement relationship is constructed; and through adjustment of a global node force vector, various planar load scenarios such as single-point, multi-point and distributed load are adapted. The application breaks through the limitation of small deformation assumption, accurately represents large deformation behavior, improves the calculation efficiency by hundreds of times compared with finite elements, controls the average error within 6%, can quantize the influence of structural parameters such as connection type, beam quantity, beam inclination angle and top angle, and provides a theoretical basis for fin effect clamp jaw engineering design.

Description

Technical Field

[0001] This invention belongs to the field of end effector / soft robot technology, and particularly relates to a force-deformation theory modeling method for fin-effect grippers based on a co-rotation model. This method, based on co-rotation theory and Newton-Raphson nonlinear iterative solution, is used to quickly predict nodal displacements under large deformation conditions and can adapt to force-deformation modeling for single-point loads, multi-point loads, and distributed loads. It can be applied to the force-deformation relationship analysis of various planar soft, rigid, and hybrid robot grippers, providing theoretical support for the structural design and performance prediction of fin-effect grippers, and is suitable for scenarios requiring adaptive grasping of geometrically complex objects, such as industrial gripping and service robots. Background Technology

[0002] Bionic fin-effect grippers have attracted widespread attention in the field of soft robotics due to their ability to passively adapt to objects with complex shapes. These grippers are typically made of flexible materials and have a fin-like structure. When subjected to external force, their contact surface and back surface undergo differentiated deformation, thereby achieving an envelope gripping of the object.

[0003] Currently, mechanical modeling of fin-effect grippers mainly faces the following technical problems and limitations:

[0004] 1. Limitations of existing theoretical models: Some studies use pseudo-rigid body models[1], chain beam constraint models[2], and Cosserrat bar theory[3] for modeling. However, these methods are usually based on the assumption of small deformation or require prior knowledge of specific stiffness matrices, making it difficult to accurately predict the nonlinear mechanical behavior of structures under large deformations. In addition, some models are only applicable to symmetric structures and have poor versatility.

[0005] 2. Limitations of existing numerical methods: Finite element analysis is currently the most commonly used and highly accurate method. However, in order to simulate complex geometric and nonlinear large deformations, finite element analysis requires extremely dense mesh generation (thousands to tens of thousands of elements) and a large number of iterative calculations, resulting in excessively long calculation time, which seriously restricts its application in scenarios with high efficiency requirements such as design iteration, parameter analysis and real-time control [4].

[0006] 3. Lack of simulation of connection methods: In existing theoretical modeling, most of them simplify the connection between the crossbeam and the front and rear side walls in the fin effect gripper structure to a fixed connection, ignoring the influence of connection methods such as hinges that may exist in the actual physical prototype on the overall structural stiffness, deformation and stability, resulting in deviations between the model and the actual situation.

[0007] Therefore, there is an urgent need for a fin effect gripper force-deformation relationship modeling method that can both ensure prediction accuracy and significantly improve computational efficiency, while also taking into account different physical connection methods. Summary of the Invention

[0008] The purpose of this invention is to provide a fin effect gripper force-deformation theory modeling method based on a co-rotation model, in order to solve the following technical problems:

[0009] 1. Overcome the shortcomings of existing fin effect gripper modeling methods that rely on small deformation assumptions, and achieve high-precision force-deformation relationship prediction in large deformation scenarios;

[0010] 2. This method addresses the issues of low computational efficiency and complex mesh generation in existing methods, significantly reducing computational costs while maintaining modeling accuracy;

[0011] 3. Decouple the axial and rotational deformation of beam elements, quantify the impact of actual structural parameters such as physical connection type on gripper performance, and improve the versatility of the model;

[0012] 4. Achieve accurate modeling of the force-deformation relationship of the fin effect gripper under various planar load scenarios such as single-point contact, multi-point contact, and distributed contact, adapting to different gripping conditions.

[0013] To solve the above-mentioned technical problems, the specific technical solution of the fin effect gripper force-deformation theory modeling method based on the co-rotation model of the present invention is as follows:

[0014] A force-deformation modeling method for fin-effect grippers based on co-rotation theory is proposed. This method decomposes the fin-effect gripper structure into several beam elements, decouples the local deformation and rigid body motion of each beam element through co-rotation theory modeling, and constructs a global force-displacement relationship using the Newton-Raphson iterative method. This achieves high-precision and high-efficiency modeling under large deformation and multi-load scenarios, and includes the following steps:

[0015] Step 1: Beam element co-rotation theory modeling: The fins, beams, and other structures of the fin effect gripper are abstracted as two-dimensional beam elements. Each beam element is anchored in the global coordinate system through two nodes. Axial deformation model and rotational deformation model are established respectively. At the same time, an effective length correction coefficient is introduced to consider the actual influence of physical hinge connections. Step 2: Global force-displacement relationship modeling: The Newton-Raphson iteration method is used to realize the force balance solution of load increment step at the global level.

[0016] Step 3: Adaptation to multi-load scenarios.

[0017] Furthermore, step 1 includes the following steps:

[0018] Step 1.1: Axial deformation modeling.

[0019] Step 1.2: Rotational deformation modeling;

[0020] Step 1.3: Constructing the tangential stiffness matrix.

[0021] Furthermore, step 1.1 includes the following steps:

[0022] Calculate the initial length based on the initial node coordinates of the beam element. and initial inclination angle When a node is displaced by an external load, the updated length is calculated. and tilt angle The axial deformation amount is obtained. Derivation of axial force based on Hooke's law ,in For elastic modulus, Let the cross-sectional area of ​​the beam element be denoted by ; an effective length correction index is introduced. ,in For node correction factors, The actual effective length of the beam element is adjusted to the hinge radius, i.e. , ; and through Euler buckling formula ,in For the moment of inertia, To achieve the equivalent length, axial force is limited to prevent beam element buckling.

[0023] Furthermore, step 1.2 includes the following steps: defining the rotation angles of the two ends of the beam element about the initial tilt axis. , The local node rotation angle is derived as follows:

[0024]

[0025] Establish the relationship between nodal bending moment and local nodal rotation angle based on standard structural analysis theory:

[0026]

[0027] in , Let be the bending moment at both ends of the beam element.

[0028] Furthermore, step 1.3 includes the following steps:

[0029] The transformation matrix is ​​derived by transforming the local and global coordinate systems. Based on the principle of virtual work equivalence, a mapping relationship between local internal forces and global nodal forces in beam elements is established. ,in For global node force vectors, The local internal force vectors are further derived to obtain the tangential stiffness matrix of the beam element. , The local stiffness matrix is:

[0030]

[0031] The tangent stiffness matrix of each element The global tangent stiffness matrix assembled into the structure :

[0032]

[0033] in, The assembly operator modifies the global tangential stiffness matrix based on the nodal support conditions, resulting in the modified global tangential stiffness matrix. The correction method is to set the rows and columns corresponding to the constrained degrees of freedom to zero.

[0034] Furthermore, step 2 includes the following steps:

[0035] Step 2.1: Total external load Divided into Each load increment step, the load of each increment step ( );

[0036] Step 2.2: Based on the beam element co-rotation modeling results, the corrected global tangential stiffness matrix is ​​used. Calculate the nodal displacement increment for each increment step. Update node displacement and node force ;

[0037] Step 2.3: Define the residual , For global node internal forces, It is also obtained from the residual, that is Set residual tolerance With a maximum number of iterations of 100, the node displacements are corrected through internal iterations until the residuals meet the tolerance or the maximum number of iterations is reached, thus completing the solution of a single load increment step.

[0038] Step 2.4: Solve all load increment steps sequentially to obtain the global force-displacement relationship of the fin effect gripper under the total external load, and realize deformation prediction under different load scenarios.

[0039] Furthermore, step 3 includes the following steps:

[0040] Based on the aforementioned global force-displacement relationship model, the external load is set as a load form for different points, and the global nodal force vector is adjusted accordingly. The position and size of non-zero elements in the model are used to model the deformation of the fin effect gripper under different planar load scenarios.

[0041] The fin effect gripper force-deformation theory modeling method based on the co-rotation model of the present invention has the following advantages:

[0042] 1. Overcoming the limitations of the small deformation assumption: By decoupling the local deformation and rigid body motion of the beam element through co-rotation theory, and combining the load increment step and the Newton-Raphson iteration method, the large deformation behavior of the fin effect gripper can be accurately characterized, solving the problem of low prediction accuracy of existing theoretical modeling methods for large deformation scenarios;

[0043] 2. Significantly improve computational efficiency: The fin effect structure is represented graphically, which greatly reduces the number of modeling units. The calculation time is only 0.035 seconds, which is hundreds of times more efficient than the tens or even hundreds of seconds of traditional finite element analysis. At the same time, it requires only a small amount of computing resources, making it suitable for the rapid design and performance prediction of grippers.

[0044] 3. High modeling accuracy: The average error between the model prediction results and the finite element analysis results is only 6%. Under different load scenarios such as single-point contact, multi-point contact, and distributed contact, the relative deviation between physical experiments and model predictions is controlled within the range of 2~6%, which can accurately reflect the actual force-deformation relationship of the fin effect gripper.

[0045] 4. High versatility: It decouples the axial and rotational deformation of beam units and, for the first time, incorporates structural parameters such as physical connection type (fixed connection, hinged connection), number of beams, beam inclination angle, and apex angle into the modeling system. It can quantify the impact of various parameters on gripper performance and is compatible with various planar load scenarios. It can be extended to the modeling of grippers for various planar soft, rigid, and hybrid robots.

[0046] 5. High engineering applicability: The model does not depend on the outline of the target object and can be applied to the grasping scenario of unknown objects. At the same time, the modeling process takes into account actual engineering factors such as hinge width and material thickness. The effective length correction improves the fit between the model and the actual physical structure, providing a reliable theoretical basis for the engineering design and optimization of fin effect grippers. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of the discretization of the beam element of the fin-effect gripper structure in this invention;

[0048] Figure 2 This is a schematic diagram of the co-rotation theory deformation of a single beam element in this invention;

[0049] Figure 3 This is a schematic diagram showing the load application positions for different physical connection types (hinged and fixed connections) in embodiments of the present invention;

[0050] Figure 4 This is a schematic diagram showing the load application positions for different structural parameters (number of beams, beam inclination angle, and apex angle) in an embodiment of the present invention. Detailed Implementation

[0051] To better understand the purpose, structure, and function of this invention, the following detailed description, in conjunction with the accompanying drawings, provides a method for modeling the force-deformation theory of fin effect grippers based on a co-rotation model.

[0052] This invention proposes a force-deformation modeling method for fin-effect grippers based on co-rotation theory. The method decomposes the fin-effect gripper structure into several beam elements, decouples the local deformation and rigid body motion of each beam element through co-rotation theory modeling, and constructs a global force-displacement relationship using the Newton-Raphson iterative method. This achieves high-precision and high-efficiency modeling under large deformation and multi-load scenarios. The specific steps are as follows:

[0053] Step 1: Modeling beam elements using the co-rotation theory

[0054] The fins, beams, and other structural elements of the fin-effect gripper are abstracted as two-dimensional beam elements. Each beam element is anchored in the global coordinate system through two nodes, such as... Figure 1 As shown, axial deformation and rotational deformation models are established respectively, and an effective length correction coefficient is introduced to consider the actual influence of the physical hinge connection:

[0055] Step 1.1: Axial deformation modeling: as shown Figure 2 As shown, the initial length is calculated based on the initial node coordinates of the beam element. and initial inclination angle When a node is displaced by an external load, the updated length is calculated. and tilt angle The axial deformation amount is obtained. Derivation of axial force based on Hooke's law ( For elastic modulus, (For the cross-sectional area of ​​the beam element); introduce the effective length correction index. ( For node correction factors, The actual effective length of the beam element is corrected to the hinge radius, i.e. , And through the Euler buckling formula ( For the moment of inertia, (For equivalent length) to limit axial force and prevent beam element buckling.

[0056] Step 1.2: Rotational Deformation Modeling: Define the rotation angles of the two ends of the beam element about the initial tilt axis. , The local node rotation angle is derived as follows:

[0057]

[0058] Establish the relationship between nodal bending moment and local nodal rotation angle based on standard structural analysis theory:

[0059]

[0060] in , Let be the bending moment at both ends of the beam element.

[0061] Step 1.3: Construction of the tangential stiffness matrix: The transformation matrix is ​​derived by transforming the local and global coordinate systems. Based on the principle of virtual work equivalence, a mapping relationship between local internal forces and global nodal forces in beam elements is established. ( For global node force vectors, (This is the local internal force vector); further derivation yields the tangential stiffness matrix of the beam element. , The local stiffness matrix is:

[0062]

[0063] The tangent stiffness matrix of each element The global tangent stiffness matrix assembled into the structure :

[0064]

[0065] in, This indicates the assembly operator. The global tangential stiffness matrix is ​​corrected based on the node support conditions (fixed hinge, rotating hinge, traversing hinge) to obtain the corrected global tangential stiffness matrix. The correction method is to set the rows and columns corresponding to the constrained degrees of freedom (such as axial displacement, tangential displacement, and rotation angle) to zero.

[0066] Step 2: Global force-displacement relationship modeling. The Newton-Raphson iterative method is used to achieve force balance solution for incremental load steps at the global level. The specific process is as follows:

[0067] Step 2.1: Total external load Divided into Each load increment step, the load of each increment step ( );

[0068] Step 2.2: Based on the beam element co-rotation modeling results, the corrected global tangential stiffness matrix is ​​used. Calculate the nodal displacement increment for each increment step. Update node displacement and node force ;

[0069] Step 2.3: Define the residual , For global node internal forces, It is also obtained from the residual, that is Set residual tolerance With a maximum number of iterations of 100, the node displacements are corrected through internal iterations until the residuals meet the tolerance or the maximum number of iterations is reached, thus completing the solution of a single load increment step.

[0070] Step 2.4: Solve all load increment steps sequentially to obtain the global force-displacement relationship of the fin effect gripper under the total external load, and realize deformation prediction under different load scenarios.

[0071] Step 3: Adaptation to Multi-Load Scenarios

[0072] Based on the aforementioned global force-displacement relationship model, the external load is set as a load form for different points, and the global nodal force vector is adjusted accordingly. By determining the position and size of non-zero elements in the model, we can model the deformation of the fin effect gripper under different planar load scenarios without requiring additional modifications to the model, thus improving its adaptability.

[0073] Example 1: Simulation Comparison of Different Physical Connection Types

[0074] 1. Simulation Settings: Using the structural parameters in Table 1, only change the connection type between the beam and the fin, and set two sets of variables:

[0075] Group 1 (Hinged): The beam can rotate freely around the connecting node without rotational constraints, and the effective length correction index is [not specified]. ;

[0076] Group 2 (Fixed Connection): The crossbeam and fins are fixedly connected without relative rotation, and the effective length correction index is [not specified]. ;

[0077] Gradient point loads were applied to different nodes (hinged group: 0.15N, 0.30N, 0.45N, 0.60N; fixed connection group: 0.25N, 0.50N, 0.75N, 1.0N), as follows: Figure 3 As shown in Figure A, the displacement of the fingertip node and the intermediate node are calculated using simulation.

[0078] Table 1: Structural parameters for different physical connection types in Example 1

[0079]

[0080] 2. Simulation results (see Table 2):

[0081] Table 2: Simulation results of Example 1:

[0082]

[0083] Where AE represents absolute error, RE represents relative error, and MAE represents the average of absolute errors. The standard deviation represents the absolute error, while the mean relative error (MRE) represents the average relative error. The standard deviation of the relative error is represented by NA, which indicates simulation model collapse. The same applies to the tables below.

[0084] Displacement characteristics: Under similar loads, the displacement of the gripper in the articulated group is greater than that in the fixed connection group, indicating that the articulated connection significantly improves the compliance of the gripper.

[0085] Modeling accuracy: The mean absolute error (MAE) and mean relative error (MRE) of the model and the finite element analysis results of this invention are as follows: Hinged group: MAE=0.36mm, MRE=4.5%, among which the error distribution of the middle nodes (No. 2 and No. 3) is uniform, at 0.29mm / 1.8% and 0.26mm / 1.4% respectively; Fixed connection group: MAE=0.28mm, MRE=5.8%, with the fingertip node (No. 4) having the best accuracy, with an average deviation of 0.115mm and a standard deviation of 0.1mm;

[0086] Load-bearing capacity: The maximum allowable load for the fixed connection group is 1.2N (without buckling), and the maximum allowable load for the hinged connection group is 0.7N. The fixed connection improves the structural stability.

[0087] 3. Conclusion:

[0088] The model of this invention can accurately quantify the impact of connection type on gripper performance. The prediction error for both connection types is controlled within 6%, which meets the engineering design requirements.

[0089] Example 2: Simulation comparison of different numbers of crossbeams

[0090] 1. Simulation settings:

[0091] Keeping other parameters in Table 1 unchanged, set the number of beams to three sets of variables: 3, 4, and 5, with corresponding load ranges of 0.15N~0.60N, 0.25N~1.0N, and 0.40N~1.6N respectively (to avoid structural buckling). Figure 4 As shown in B, the deformation distribution and load-bearing characteristics of the gripper are simulated and analyzed under different numbers of crossbeams.

[0092] 2. Simulation results (see Table 3):

[0093] Table 3: Simulation results of Example 2:

[0094]

[0095] Deformation distribution: Regardless of the number of crossbeams, the central area nodes (No. 2 and No. 3) have the largest deformation, while the edge nodes have the smallest deformation. The more crossbeams there are, the stronger the adaptability of the central area and the higher the fingertip stiffness.

[0096] Modeling accuracy: The average deviations between the model predictions and finite element analysis for the three sets of variables are as follows: 3-beam group: MAE=0.35mm, MRE=6.6%, fingertip node deviation 0.65mm / 8.7%; 4-beam group: MAE=0.28mm, MRE=5.8%, the middle node (No. 3) has the smallest error, at 0.04mm / 0.6%; 5-beam group: MAE=0.37mm, MRE=6.3%, all node errors are evenly distributed, with no obvious concentrated deviation areas;

[0097] Computational efficiency: The model of this invention requires only 28 to 34 beam elements and takes 0.035 to 0.045 seconds to compute, while finite element analysis requires 6295 to 6630 meshes and takes 12.1 to 15.3 seconds to compute, which is more than 300 times more efficient.

[0098] 3. Conclusion:

[0099] The model of this invention can accurately capture the influence of the number of crossbeams on the stiffness and load-bearing capacity of the gripper, with an error of around 6%, providing a precise basis for the selection and design of the number of crossbeams.

[0100] Example 3: Simulation comparison of different beam inclination angles

[0101] 1. Simulation settings:

[0102] Keeping other parameters in Table 1 unchanged, set the beam inclination angle to three sets of variables: -10°, 0°, and +10°, and apply a gradient point load of 0.25N~1.0N. Figure 4 As shown in Figure C, the effect of the tilt angle on the deformation characteristics of the gripper is analyzed through simulation.

[0103] 2. Simulation results (see Table 4):

[0104] Table 4: Simulation results of Example 3:

[0105]

[0106] Stability characteristics: The maximum unstable loads (before buckling) are 1.15N for the -10° group, 1.2N for the 0° group, and 1.35N for the +10° group. The bearing capacity of the +10° tilt group is 14.5% higher than that of the -10° group.

[0107] Modeling accuracy: The average deviations of the three groups of variables are as follows: -10° group: MAE=0.23mm, MRE=4.6%, with the bottom node (No. 1) having the largest error (0.34mm / 9.3%); 0° group: MAE=0.28mm, MRE=5.8%, with uniform error distribution across all nodes and no significant deviation; +10° group: MAE=0.23mm, MRE=4.5%, with the middle node (No. 3) having the best accuracy (0.12mm / 1.6%).

[0108] 3. Conclusion:

[0109] The model of this invention can effectively characterize the coupled effect of beam tilt angle on gripper compliance and stability. The prediction error at different tilt angles is less than 6%, providing theoretical support for tilt angle parameter optimization.

[0110] Example 4: Simulation comparison of different vertex angles

[0111] 1. Simulation settings:

[0112] Keeping other parameters in Table 1 unchanged, set three sets of variables for the apex angle to 20°, 30°, and 40°, corresponding to maximum applied loads of 1.0N, 1.2N, and 1.4N respectively. Figure 4 (D) Simulation analysis of the influence of the apex angle on the deformation range and structural dimensions of the gripper.

[0113] 2. Simulation results (see Table 5):

[0114] Table 5: Simulation results of Example 4:

[0115]

[0116] Deformation characteristics: The larger the apex angle, the greater the deformation in the central area (nodes 2 and 3), and the smaller the deformation at the fingertip nodes. The displacement of the central node in the 40° apex angle group is about 60% higher than that in the 20° group (under a 1.4N load, the displacement of node 2 in the 40° group is 2.18mm, while the displacement of node 2 in the 20° group is 0.85mm under a 1.0N load).

[0117] Load-bearing capacity: The maximum permissible load increases with the apex angle: 1N for 20° group, 1.2N for 30° group, and 1.4N for 40° group. For every 10° increase in apex angle, the load-bearing capacity increases by approximately 20%.

[0118] Modeling accuracy: The deviations between the model predictions and finite element analysis for the three groups of variables are as follows: 20° group: MAE=0.28mm, MRE=5.8%, fingertip node error 0.19mm / 6.0%; 30° group: MAE=0.38mm, MRE=4.3%, center node (No. 2) error 0.83mm / 6.4%; 40° group: MAE=0.63mm, MRE=5.2%, fingertip node error is the smallest, at 0.07mm / 3.6%.

[0119] Structural impact: The increased apex angle leads to an increase in the overall size and weight of the gripper. The gripper width of the 40° group is about 25% greater than that of the 20° group. The model of this invention can make comprehensive predictions by combining structural dimensions and performance.

[0120] 3. Conclusion:

[0121] The model of this invention can accurately reflect the comprehensive influence of the apex angle on the gripper's compliance, load-bearing capacity and structural dimensions. The average error under different apex angles is controlled within 5.8%, with the center node (No. 3) having the best accuracy (deviation ≤ 2.9%), meeting the design requirements of precision gripping scenarios.

[0122] It is understood that the present invention has been described through some embodiments, and those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of the present invention.

Claims

1. A fin effect gripper force-deformation theory modeling method based on co-spin theory, characterized in that, The method decomposes the fin-effect gripper structure into several beam elements, decouples the local deformation and rigid body motion of each beam element through co-rotation theory modeling, and constructs a global force-displacement relationship by combining the Newton-Raphson iteration method, achieving high-precision and high-efficiency modeling under large deformation and multi-load scenarios, including the following steps: Step 1: Beam element co-rotation theory modeling: The fins, crossbeams and other structures of the fin effect gripper are abstracted into two-dimensional beam elements. Each beam element is anchored in the global coordinate system through two nodes. Axial deformation model and rotational deformation model are established respectively. At the same time, an effective length correction coefficient is introduced to consider the actual influence of physical hinge connection. Step 2: Global force-displacement relationship modeling: The Newton-Raphson iterative method is used to achieve force balance solution of load increment step at the global level; Step 3: Adaptation to multi-load scenarios.

2. The fin effect gripper force-deformation theory modeling method based on co-rotation theory according to claim 1, characterized in that, Step 1 includes the following steps: Step 1.1: Axial deformation modeling; Step 1.2: Rotational deformation modeling; Step 1.3: Constructing the tangential stiffness matrix.

3. The fin effect gripper force-deformation theory modeling method based on co-rotation theory according to claim 2, characterized in that, Step 1.1 includes the following steps: Calculate the initial length based on the initial node coordinates of the beam element. and initial inclination angle When a node is displaced by an external load, the updated length is calculated. and tilt angle The axial deformation amount is obtained. Derivation of axial force based on Hooke's law ,in For elastic modulus, Let the cross-sectional area of ​​the beam element be denoted by ; an effective length correction index is introduced. ,in For node correction factors, The actual effective length of the beam element is adjusted to the hinge radius, i.e. , ; And through the Euler buckling formula ,in For the moment of inertia, To achieve the equivalent length, axial force is limited to prevent beam element buckling.

4. The fin effect gripper force-deformation theory modeling method based on co-rotation theory according to claim 2, characterized in that, Step 1.2 includes the following steps: defining the rotation angles of the two ends of the beam element about the initial tilt axis. , The local node rotation angle is derived as follows: ， Establish the relationship between nodal bending moment and local nodal rotation angle based on standard structural analysis theory: ， in , Let be the bending moment at both ends of the beam element.

5. The fin effect gripper force-deformation theory modeling method based on co-rotation theory according to claim 2, characterized in that, Step 1.3 includes the following steps: The transformation matrix is ​​derived by transforming the local and global coordinate systems. Based on the principle of virtual work equivalence, a mapping relationship between local internal forces and global nodal forces in beam elements is established. ,in For global node force vectors, The local internal force vectors are further derived to obtain the tangential stiffness matrix of the beam element. , The local stiffness matrix is: ， The tangent stiffness matrix of each element The global tangent stiffness matrix assembled into the structure : ， in, The assembly operator modifies the global tangential stiffness matrix based on the nodal support conditions, resulting in the modified global tangential stiffness matrix. The correction method is to set the rows and columns corresponding to the constrained degrees of freedom to zero.

6. The fin effect gripper force-deformation theory modeling method based on co-rotation theory according to claim 1, characterized in that, Step 2 includes the following steps: Step 2.1: Total external load Divided into Each load increment step, the load of each increment step ( ); Step 2.2: Based on the beam element co-rotation modeling results, the corrected global tangential stiffness matrix is ​​used. Calculate the nodal displacement increment for each increment step. Update node displacement and node force ; Step 2.3: Define the residual , For global node internal forces, It is also obtained from the residual, that is Set residual tolerance With a maximum number of iterations of 100, the node displacements are corrected through internal iterations until the residuals meet the tolerance or the maximum number of iterations is reached, thus completing the solution of a single load increment step. Step 2.4: Solve all load increment steps sequentially to obtain the global force-displacement relationship of the fin effect gripper under the total external load, and realize deformation prediction under different load scenarios.

7. The fin effect gripper force-deformation theory modeling method based on co-rotation theory according to claim 1, characterized in that, Step 3 includes the following steps: Based on the aforementioned global force-displacement relationship model, the external load is set as a load form for different points, and the global nodal force vector is adjusted accordingly. The position and size of non-zero elements in the model are used to model the deformation of the fin effect gripper under different planar load scenarios.

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