TWI857434B - Topology optimization method for design of compliant constant-force mechanisms, end-effectors, computer program products and computer readable recording medium for designing such - Google Patents

Abstract

The invention provides a topology optimization method for design of compliant constant-force mechanisms, comprising a topology optimization algorithm that uses a composite objective function considering the actual output displacement and artificial output force. A morphology-based method is developed to ensure the connectivity of topological results. The topology optimization method can be used to develop innovative compliant constant-force mechanisms that provide constant output force during operation.

Description

Topology-optimized flexible isomechanical mechanism, end effector and design method thereof, computer program product, and computer-readable recording medium

The present invention relates to an isomechanical mechanism and an end effector that can be used in the automation industry (such as an automation system and a robot arm), and in particular to a flexible isomechanical mechanism and an end effector that are designed with topology optimization and a design method thereof.

Robotic arms are increasingly used in industry. Although their accuracy is constantly improving, their working stability is still prone to slight changes due to various uncertain factors such as the production environment, component size errors, rigidity, positioning, etc.

In order to improve the stability of force output during robot processing (such as applied to polishing, machining, etc.), some robots are equipped with a flexible constant-force mechanism, which can achieve an effect of approximate constant-force output within a specific input range through an integrated structural deformation. It has the advantages of simple structure, zero backlash, zero noise, light weight, and no need for lubrication.

However, most of the existing flexible isostatic mechanisms are modified based on the existing type, and it is difficult to break away from the existing structural type. Furthermore, the performance of the existing flexible isostatic mechanisms still needs to be improved. Specifically, the isostatic value still has the problem of excessive force amplitude, resulting in unsatisfactory actual isostatic effect of the mechanism. Secondly, the isostatic interval is too small relative to the length of the mechanism, resulting in poor size efficiency of the overall mechanism.

Therefore, the inventors propose a design method of a flexible isomechanical mechanism with topological optimization design to break through the structure of the existing flexible isomechanical mechanism, wherein the target function adopts a composite target function that considers the real output displacement and the virtual output force.

Structural analysis can be performed using the geometric nonlinear finite element method, and hyperelastic body elements can be added to avoid element distortion during large deformations. Design variables can be updated using the moving asymptote method.

In order to solve the problem of chessboard grid and grid dependency, a density filtering algorithm can be further added to the topology optimization process, and the problem of too many gray-level elements after filtering can be improved through a parametric projection method, and the original design variables can be converted into solid density as parameters for finite element analysis.

Furthermore, to ensure the connectivity and integrity of the topologically optimized structure, an update process can be designed based on morphological calculations to update the design variables and entity density at the end of each iteration to avoid single-point connections or unconnected structures, while reducing small holes and branch structures in the structure.

The above design method can be constructed as a program in practice and stored in a computer program product or a computer-readable recording medium. When the computer loads and executes the program, it can be implemented accordingly.

The present invention is a flexible isomechanical mechanism with topology optimization design, which is manufactured by the design method of the flexible isomechanical mechanism with topology optimization design described above and adopts flexible materials. The flexible isomechanical mechanism with topology optimization design includes two opposite ends, and the two ends can be respectively connected to a robot arm end fixture and a tool end fixture (for example, for installing grinding, polishing, and other processing tools) to form an end effector.

Through the above features, the following effects can be achieved:

1. Using topology optimization method to design flexible isotropic mechanisms does not require designing based on existing mechanisms, which is conducive to breaking through the structure of existing flexible isotropic mechanisms.

2. The morphological-based update process can ensure the integrity and connectivity of the topology structure, avoid single-point connection and structural disconnection in the topology optimization results, and make the topology optimization iterative process stable and convergent.

3. By adding superelastic elements to low solid density element areas, element distortion can be avoided during large deformations.

4. The results of static compression test and dynamic curved surface contact test show that the flexible isomechanical mechanism and end effector have good isomechanical range and isomechanical effect.

In combination with the above technical features, the main functions of the flexible isomechanical mechanism, end effector and design method thereof, computer program product, and computer-readable recording medium of the topology optimization design of the present invention can be clearly presented in the following embodiments with accompanying drawings. It should be noted that, for ease of understanding, similar functional components in each drawing will use similar or identical component symbols.

The design method of the flexible isotropic mechanism of the topology optimization design of the embodiment of the present invention can be constructed as a program and stored in a computer program product or a computer readable recording medium. When the computer loads the program and executes it, the design method of the flexible isotropic mechanism of the topology optimization design can be completed.

The embodiment of the present invention uses a topology optimization method to design a flexible isotropic mechanism. Compared with other flexible mechanism design methods, its advantage is that it does not need to be designed based on an existing mechanism, which is conducive to innovative mechanism design. This embodiment mainly adds a morphology-based update process to the topology optimization process to ensure the connectivity and integrity of the topology optimization results, and uses the SIMP (solid isotropic material with penaliztion, SIMP) method as the assumption of topology optimization design variables, and uses the MMA method (Method of Moving Asymptotes, MMA) to update the design variables.

Please refer to FIG. 1, which is a main flow chart of topology optimization in this embodiment. The steps are as follows:

Step 1: Define the design interval, boundary conditions, design parameters and initial values according to requirements.

Step 2: Perform finite element analysis on the solid density according to the set boundary conditions.

Step 3: Calculate element sensitivity using finite element analysis results.

Step 4: Update the design variables using the MMA method.

Step 5: Update the design variables using the filtering algorithm.

Step 6: Use the projection method to obtain the entity density.

Step 7: Use the morphology update process to update the design variables and entity density.

Step 8: Determine whether convergence is achieved. If convergence conditions are met, the topology optimization is terminated. If not, repeat step 2 to step 7 until convergence conditions are met.

It takes a long time to perform nonlinear linear element analysis, so this embodiment uses a two-dimensional topology optimization method to simplify the problem. As shown in FIG2 , the design interval is discretized into finite elements, each element is, for example, a square element with equal length and width, nelx is the number of elements in the x direction of the design interval, and nely is the number of elements in the y direction of the design interval.

Each element has its corresponding element density x _i , which is used as the design variable for topological optimization, where the subscript i represents the i - th element in the design interval, and the design variable is a value between 0 and 1. A value of 1 is a solid element (indicating the existence of a solid structure), and a value of 0 is an empty element (indicating that the structure is deleted). In order to avoid the rigid matrix with singularity when the design variable is 0, which makes the optimization result difficult to converge. In this embodiment, the design variable is a value between x _min and 1, where x _min is an extremely small value greater than 0, for example, it can be set to 0.001. When the design variable approaches x _min, it is regarded as a structure to be deleted, otherwise it is regarded as a solid structure when it approaches 1.

The following is a detailed introduction to each process of the design method of the embodiment of the present invention, which mainly includes: 1. finite element analysis, 2. target function setting, 3. hyperelastic body assumption method, 4. element sensitivity calculation, 5. variable update method (MMA), 6. filtering and projection method and morphology-based update method, and finally a complete description of 7. topology optimization process of flexible isomechanical mechanism.

1. Finite element analysis:

Finite element analysis can be mainly divided into two types: linear and nonlinear. In linear finite element analysis, the stress and strain curves are linear, and the force and displacement curves are also linear, which can be expressed as formula (A-1). Nonlinear finite element analysis can be further divided into three categories, namely material nonlinearity, geometric nonlinearity and boundary nonlinearity. Material nonlinearity is caused by the nonlinearity of the stress-strain curve of the material itself, such as hyperelastic materials; geometric nonlinearity is when the structure produces large displacement and large deformation, its global rigidity matrix will be dependent on the displacement vector, which can be expressed as formula (A-2); boundary nonlinearity is that the force state of the structure will change with the displacement during the analysis, that is, the boundary condition is nonlinear, such as contact problems.

Formula (A-1): F = KU;

Formula (A-2): F = K(U)U;

The flexible isotropic mechanism of this embodiment must achieve isotropic output by buckling the mechanism itself, and the actuation process is usually accompanied by large displacement and large deformation. Therefore, assuming that the flexible isotropic mechanism is a geometric nonlinear problem with large displacement and small strain, the finite element method of geometric nonlinearity cannot be solved directly through the inverse matrix, and needs to be gradually approximated through superposition to finally obtain a solution when the structure is approximately balanced.

Based on the assumption of large displacement and small strain, the material can be assumed to be linear and the strain tensor can be defined using the Green-Lagrange strain. :

(A-3): ;

in Represents displacement in the i direction Partially differentiate in the j direction.

Define B as the transformation matrix that converts displacement changes to strain changes for subsequent use:

(A-4): .

Using Hooke's law to calculate the second Piola-Kirchhoff stress, the stress tensor can be expressed as:

(A-5): ;in Represents the constitutive tensor of element e.

Formula (A-5) can be rewritten as formula (A-6) after combining with SIMP method:

Formula (A-6): ; where _xe is the element density of element e , p is the penalty coefficient, is the constitutive tensor when x _e = 1.

The error in force balance is defined as the nonlinear balance residual ( R ):

Formula (A-7): ; Where F _ext is the one-dimensional external force vector, and s is the internal stress (second Piola-Kirchhoff stress) vector.

Equation (A-7) is the equilibrium equation of the nonlinear finite element method. When the residual vector is equal to the zero vector, it is in equilibrium. It can be solved by the Newton-Raphson method. The formula calculation can refer to equations (A-8), (A-9) and (A-10).

Formula (A-8): ;

Formula (A-9): ;

Formula (A-10): ;

in is the external force vector, represents the internal force vector at the it-th iteration, represents the tangent stiffness matrix at the it - th iteration, which can be regarded as the slope of the force-displacement curve during the Newton-Larsen method iteration process, see equation (A-10). is the displacement increment at the it -th iteration, used to obtain the displacement value at the next iteration When R = 0, the Newton-Larsen method ends the iteration, while the general nonlinear finite element method uses the Euclidean norms of displacement or force as the convergence criterion during calculation, and ends the iteration when its value is less than the allowable error.

2. Design of virtual spring:

When defining the boundary conditions for topology optimization, virtual springs are generally added to the input and output ends to simulate the rigidity of the input and output contact objects of the actuator. The virtual springs are established by merging the rigidity matrix K _S of the virtual springs into the global rigidity matrix K. After adding the virtual springs, the topological structure can be promoted to generate an appearance that is conducive to force transmission and load resistance. Please refer to Figures 3 and 4 for the model with virtual springs, where Ω is the design interval, Γ _d is the boundary condition, u _in and u _out represent the input displacement and output displacement respectively, k _in and k _out represent the spring constants at the input and output ends respectively, and _Fin and F _out represent the input force and output force respectively.

The virtual spring must be consistent with the input and output directions when actuated, that is, the virtual spring cannot produce tangential deformation. If tangential deformation occurs, it means that lateral force is generated, which may lead to the topology result being not as expected, or even convergence failure. The following methods can be used to avoid or reduce the influence of lateral force:

Method 1: Under the symmetrical boundary condition, the virtual spring is fixed at the two symmetrical output ends of the whole model, and the virtual spring is perpendicular to the symmetry axis, as shown in Figure 5.

Method 2: Under the symmetrical boundary condition, fix the virtual spring on the output end of the half-model symmetrical axis, and the virtual spring is in the same direction as the alignment axis, as shown in Figure 6.

Method 3: Fix the virtual spring far away from the output end of the full model, as shown in Figure 7. Represents boundary conditions at extreme distances.

Among the above methods, method 1 and method 2 require symmetrical boundary conditions when used to avoid the influence of lateral force, but they have limitations in use. Method 3 is an approximate method that can reduce the influence of lateral force and is not restricted by symmetrical boundary conditions. This method maximizes the original length of the virtual spring so that the lateral deformation is relatively close to zero. The calculation is as shown in formula (A-11). The deformation diagram can be referred to in Figure 8.

Formula (A-11): ;

Where F _S is the virtual output force, F _S,n and F _S,t are the virtual output forces in the normal and tangential directions, l ₀ is the original length of the virtual spring, l is the length of the virtual spring after deformation, Δ u _n and Δ u _t are the displacements of the virtual spring in the normal and tangential directions.

2. Objective function setting (optimization problem of minimizing the sum of square errors of the composite objective function):

The model assumptions of topology optimization can be divided into virtual models with virtual springs added and real models without virtual springs, as shown in Figures 9 and 10, where the superscript ^ is the variable definition of the real model. Generally, virtual springs are added during topology optimization to stabilize the finite element analysis results and assist in topology optimization.

In the topological optimization problem of the flexible isomechanical mechanism, this embodiment adds a virtual spring at the output end, uses the virtual spring constant and the virtual output displacement to represent the virtual output force, and solves it through the error minimization problem. FIG11 is a relationship diagram between the input displacement and the output force of the original flexible mechanism, and FIG12 is a relationship diagram between the input displacement and the output force of the flexible isomechanical mechanism, where F _obj is the target output force, u _in,pn is the input displacement under the pnth input process, F _out,pn is the output force under the pnth input process, It is the output force error under the pnth input process.

If only the minimization of the virtual output force error is considered, two phenomena are likely to occur. The first is that both the virtual output displacement and the real output displacement tend to be close to zero, which will cause the flexible isomechanical mechanism to have almost no output displacement when operating under no load. The second is that the real output displacement changes continuously during the topological optimization iteration process, resulting in the inability to converge the results. Therefore, the optimization problem of minimizing the sum of square errors of the composite objective function that considers both the real output displacement and the virtual output force is adopted here. The definition is as follows:

Formula (A-12): Objective function: ; Restricted: ; ; ; ; Error formula: ; ;

Where _hi is the error formula of the objective function, R is a nonlinear equilibrium residual under a virtual model, U is a global displacement vector under the virtual model, is a nonlinear equilibrium residual under a real model, is a global displacement vector under the real model, _Ne represents the number of elements, is the element density x _i calculated by a density filtering evolution method and a parameterized projection method, _vi is the volume of the i-th element in the design interval, V ^* is a target volume, x _min is a preset minimum value of the design variable, h ₁ is the error formula of the true output displacement, represents the actual output displacement error under the pnth input process. The subscript pn represents the number of different input processes. represents the actual output displacement, LimitL represents the target value of the actual output displacement, is the virtual output force error under the pn -th input process, F _out,pn is the output force under the pn -th input process, and F _obj is the target output force.

In this embodiment, an initial target force F _obj is defined before topology optimization. This is a guess value. The optimization goal is to minimize the error between the output force and the target force at the specified input displacement. When the change of the virtual output force converges in the inner loop, it will enter the outer loop to check whether it meets the target force. If it meets the target force, the optimization ends. If not, the target force is corrected and iterated again. This method can obtain the result of equal force output by correcting the target force and re-optimizing when the topology optimization result cannot reach the target force. The formula for correcting the target force is as follows:

Formula (A-13): ;

Where tp is the sum of the numbers ( pn ) of different input processes.

In this embodiment, the topology optimization process is divided into inner and outer loops, wherein the convergence condition of the inner loop is that the variation of the virtual output force is less than the allowable value and the projection parameter in the following formula (A-49) is Set to 512. Since the virtual output force of the flexible isotropic mechanism is prone to large variations between generations during topology optimization, this embodiment considers the virtual output force of the current generation and the previous nine generations to obtain a more stable topology result. The calculation method is as follows:

Formula (A-14): ;

Where F _tol is the inner loop convergence error tolerance, iter is the number of iterations of the inner loop. The convergence condition of the outer loop is the force error under each input process. Is it less than the force error tolerance errtol ?

3. Hyperelastic body assumption method:

When using the elements assumed by the SIMP method to perform finite element analysis, the low-density element region is prone to distortion. This phenomenon will cause the finite element analysis to fail, thereby causing the topology optimization process to be interrupted. Therefore, this embodiment can further use the hyperelastic body assumption method to solve the problem of low-density element regions being prone to distortion.

As shown in Figure 13, the hyperelastic body assumption method means adding hyperelastic body materials to the areas with lower element density in the original finite element model, while not adding them to the areas with higher element density, so that they share nodes with low-density elements and medium-density elements, that is, superimposing hyperelastic body elements on the original elements to avoid distortion. Here, it is assumed that the hyperelastic body material is an isotropic material and the Yeoh model is selected as the hyperelastic body model. Its strain energy function is as follows:

Formula (A-15): ;

in is the strain energy function, represents the first strain invariant of the Green strain tensor. is the material parameter of the i- th hyperelastic body.

In Yeoh's model and The parameter provides similar effect to the increase of stress at large deformation, so The parameters are set to 0 and combined with the concept of SIMP method, the elastic tensor of the original elastic material and the strain energy function of the specified hyperelastic material (A-15) are converted into equations (A-16) and (A-17):

Formula (A-16): ;

Formula (A-17): ;

in represents the linear elastic tensor of a single element, represents the linear elastic tensor of the solid element, represents the strain energy function of a single element, represents the strain energy function of a solid element (element density is 1), is the material parameter of the ith hyperelastic body of a single element.

If there is local instability in the structure under small strain conditions, its impact on the results is relatively small, so The value of the parameter should be set to a minimum value, which can be referred to in formula (A-18); in the case of large strain, the stiffness of the hyperelastic element is The parameters are highly dependent, the larger Parameters can effectively reduce structural instability, but too large However, if the parameters are too large, the hyperelastic body may become too rigid, which may cause distortion of the simulation results. The parameter is an appropriate value. Set an equivalent strain valve value here. , and compare it with the average equivalent strain of the element. If the ratio is greater than 1, it means that the element is unstable and may be distorted, so this is used as an adjustment. The basis of the parameters can be found in formula (A-19).

Formula (A-18): ;

Formula (A-19): ;

in is the Young's modulus of the original elastic material, is the average equivalent strain of a single element (the average von Mises strain).

4. Element sensitivity calculation:

Due to the actuation characteristics of the flexible isomechanical mechanism, this embodiment uses the geometric nonlinear finite element method for analysis, and uses the APDL (ANSYS Parametric Design Language) of the commercial software ANSYS to obtain the results of the geometric nonlinear analysis.

Here, in the boundary condition setting of the flexible isomechanical mechanism, a single-point output method is adopted. A virtual spring is added to the output end to simulate the rigidity of the flexible isomechanical mechanism when it contacts the object, and the virtual output displacement is used. and virtual spring constant To calculate the virtual output , the relationship is as follows:

Formula (A-20): ;

This embodiment uses filtering algorithms and projection methods in the topology optimization process. If the original design variable is , then define the design variable after the filtering algorithm is calculated as the filtering variable , and the filter variable is the entity density after the projection function operation Element sensitivity is defined as the partial derivative of the target function with respect to the design variables, as shown in formula (A-21). Therefore, the conversion between the above variables must be considered when calculating element sensitivity.

Formula (A-21): ;

in represents the target function, For The elemental sensitivity of each element.

Through ANSYS analysis and transformation of formula (A-20), the information required for calculating element sensitivity including external force vector, internal force vector and global rigidity matrix is obtained. Then, according to the MMA theory described later, the original complex objective function formula (A-12) is transformed into the constraint formula of the MMA general formula. , , and , while element sensitivity The corresponding element sensitivity is also changed to MMA restricted , , and .

According to the assumptions of the physical model and the virtual model, the element sensitivity is divided into two categories for calculation, and the MMA constraint h ₁ is defined as , and h2 , h3 and h4 _are defined as h , and the element sensitivity of _{the MMA} restriction Defined as and and Defined as , and then the two MMA restricted styles ( and ) for solid density After partial differentiation and transformation by equation (A-20), the physical element sensitivity of the MMA constraint under the virtual model and the real model can be obtained:

Formula (A-22): ;

The analytical solution of element sensitivity can be divided into two solutions: direct method and adjoint method. In order to simplify the calculation and quickly derive the element sensitivity, this embodiment adopts the adjoint method for solution. The virtual output displacement and the real output displacement can be expressed by the adjoint method as follows:

Formula (A-23): ;

Where L ^T is the accompanying load vector, the degree of freedom of this vector in the output direction is 1, and the degrees of freedom in other directions are all 0, so that the target displacement is changed from the global displacement vector U ( ) was proposed in .

Since the nonlinear residual term of the structure is zero after nonlinear analysis equilibrium, the product of the Lagrange multiplier and the nonlinear residual term (zero term) can be added to equation (A-23) to rewrite the output displacement as:

Formula (A-24): ;

in and are the Lagrange multipliers in the virtual and real models, and They are the nonlinear equilibrium residual terms under the virtual and real models respectively.

Formula (A-24) is valid after the nonlinear analysis equilibrium of the structure, and can be combined with formula (A-10) to transform formula (A-22) into:

Formula (A-25): ;

The Lagrange multipliers in the virtual and real models can be assumed to be:

Formula (A-26): ;

By using the assumption of equation (A-26), the coefficients of the partial differential of the global displacement matrix in equation (A-25) can be made zero. After simplification, the following equation can be obtained:

Formula (A-27): ;

Under the assumption of hyperelastic body, the elements are divided into original materials and hyperelastic body materials, and the nonlinear residual term of equation (A-7) can be rewritten as:

Formula (A-28): ;

in and They are the one-dimensional external force vectors in the virtual model and the real model, and They are the element internal force vectors of the original elastic material in the virtual model and the real model, and They are the element internal force vectors of the hyperelastic material in the virtual model and the real model respectively.

Substituting formula (A-28) into formula (A-27), we can get:

Formula (A-29): ;

The partial differentials of the internal force vectors of the original elastic material and the hyperelastic material with respect to the solid density can be converted into a convenient form for calculation through the tensor concept formula (A-16) and formula (A-17) under the SIMP method:

Formula (A-30): ;

Formula (A-31): ;

Substituting equations (A-30) and (A-31) into equation (A-29), we can obtain the physical element sensitivity of the MMA constraint:

Formula (A-32): ;

The coefficient of the internal force term of the hyperelastic body element is added with 10 ^-10 in order to avoid the situation where the denominator is zero when the solid density is 1.

5. Variable Update Method (MMA):

Since the objective function of this embodiment is relatively complex, the moving asymptotes (MMA) method can be used as a variable update method. MMA is applicable to optimization with multiple variables and multiple constraints. By approximating the objective function obtained under the current design variables, the problem is converted into a separable convex function sub-problem, which is then solved by the dual method. The general formula of the optimization problem is as follows:

Formula (A-33): Objective function: ; Restricted: ; ;

in ,Right now is the design variable vector and satisfies all design variables _xi is a real number between xmin and 1, _f0 is the _original objective function, is restricted, and exist where must be a continuously differentiable function. , , and is the initial constant of MMA, which must satisfy , , , and And when Hour y ( ) and z are artificial optimization parameters of MMA, which can make MMA more flexible in applying to different optimization problems, such as minimizing the sum of squares and minimizing the maximum value.

The solution process of MMA method in topology optimization of flexible isomechanical mechanism is as follows:

Step 1: Set MMA parameters and given design variables And define the initial iteration number iter = 0;

Step 2: Use finite element analysis and element sensitivity analysis to calculate and the gradient value To generate an approximate function ;

Step 3: From the approximate function Generate a sub-optimization problem ^Piter and solve the sub-optimization problem ^Piter by dual method ^;

Step 4: Optimization Problem The best solution is used as the design variable for the next iteration. , and set iter = iter + 1 and return to step 2;

The sub-optimization problem of equation (A-33) can be expressed as the following equation (A-34): Objective function: ; Restricted: ; ; ; ;

in is the lower limit, is the upper limit.

Approximate functions for suboptimization problems for:

Formula (A-35): ;

Formula (A-36): ;

Formula (A-37): ;

Formula (A-38): ;

Formula (A-39): ;

Formula (A-40): ;

The upper and lower bounds in formula (A-34) are defined as:

Formula (A-41): ;

Formula (A-42): ;

in is the lower boundary of the moving asymptote, is the upper boundary of the moving asymptote, move is an adjustable MMA sub-problem parameter, and the upper and lower boundaries of the moving asymptote are defined after the first two iterations and the subsequent iterations, as shown in equations (A-43) and (A-44), respectively. Asyinit , asydecr , and asyincr are all adjustable MMA sub-problem parameters:

When iter < 3:

Formula (A-43): ;

when :

Formula (A-44): ;

Formula (A-45): ;

The general formula of the MMA minimization sum of squares problem is selected and applied to the optimization problem of this embodiment. The MMA problem parameters in the original MMA general formula of formula (A-33) are adjusted according to Table 1 and converted into the minimization sum of squares problem of formula (A-46):

Formula (A-46): Objective function: ; Restricted: ;

The constraints in formula (A-46) represent the objective function and constraints in formula (A-12) respectively: represents the original multi-criteria objective function, i.e. , , and ,and , represents the number of original target functions, represents the number of constraints in the original problem, represents the original volume rate constraint. In addition, in this embodiment, some parameters in the MMA sub-problem are adjusted, and the parameter settings can refer to Table 2.

Table 1: MMA parameter setting table: Restricted Parameters Target function parameters m = 2 p ^* + q ^*

Table 2: MMA sub-problem parameter setting table: Original parameter value Modified parameter value move 0.5 0.2 asyinit 0.5 0.5 asydecr 0.7 0.65 asyincr 1.2 1.05

6. Filtering and projection methods and morphology-based updating methods:

There are two numerical problems that are easy to encounter in the topology optimization process, namely checkerboard and mesh dependence. The main reason for the checkerboard is that the rigidity required in the area during numerical analysis is smaller, which causes the topology optimization result to produce a structure distribution similar to a chessboard. This phenomenon will make the topology optimization result unmanufacturable; mesh dependence means that when topology optimization is performed with meshes of different sizes, the results produced have different structures.

In order to ensure the manufacturability and mesh independence of the structure, a filtering algorithm can be used in the topology optimization based on the SIMP method. This method is mainly divided into two categories: element sensitivity filtering algorithm (sensitivity filter scheme) and density filtering algorithm (density filter scheme). The density filtering algorithm is used here.

The density filtering algorithm is to calculate the weighted average of the design variables of element i and the surrounding elements contained in the filter radius r to strengthen the correlation between elements, as shown in Figure 14, where ri _,j is the center distance from element i to element j . The calculation formula can refer to formula (A-47), and formula (A-48) is its weight, which is a linear attenuation function (conical function). The closer the distance to element i is, the greater the weight value is, and vice versa.

Formula (A-47): ;

Formula (A-48): ;

in represents the filtered variable after filtering the i- th design variable. is the set of elements within the filter radius of the i - th element, is the weight function, _vj is the volume of the jth element, r is the filter radius, and They are the center coordinates of the i - th element and the center coordinates of the j- th element respectively.

After using the filtering algorithm, it is easy to generate transitional grayscale elements (element density between 0 and 1) between solid elements and empty elements, so it is difficult to define whether the element exists. Usually, contour maps are used to obtain the structural outline during manufacturing. In order to avoid the difficulty in defining the structure, a projection method can be added after the density filtering algorithm to improve the situation of too many grayscale elements.

The projection method is a method for quickly binarizing a value. The parametric projection method used in this embodiment is a combination of the concepts of dilation projection and erosion projection, which can alleviate the poor convergence problem caused by the excessive volume change after projection of the above two methods. The parametric projection method formula is:

Formula (A-49): ;

in For the The physical density after the projection of the filter variables. is the projection parameter, The projected valve value can be greater than the valve value after projection. The filter variable tends to be close to 1, and those less than the threshold value tend to be close to 0, and The value is positively correlated with the change in projection, as shown in Figure 15. Different times The projection result produced by the value is When the result is close to binary, the result can be obtained. In this embodiment, the filtered variable after the projection function operation is called entity density, and the entity density is used to replace the original design variable to perform nonlinear finite element analysis. The threshold value h of the parametric projection method used in this embodiment is 0.5, and the projection parameter b starts from the initial value 1. When the number of iterations reaches 50 or the force variation converges, it is multiplied by 2 until the projection parameter b is equal to 512.

According to the definitions of equations (A-47) and (A-49), the solid density is a function of the filter variable, and the filter variable is a function of the design variable. Therefore, the design variable has an effect on the MMA constraint ( and ) and It can be expressed as:

Formula (A-50): ;

in and The definition can be found in formula (A-32).

Combining equations (A-47), (A-49), (A-50) and (A-32), the composite objective function proposed in this embodiment can be converted into an independent MMA constraint equation ( and ) with respect to the design variables, the complete expansion is shown below:

Formula (A-51): ;

The internal force vector ( )and ( ) can be directly obtained from the ANSYS finite element analysis results, and the Lagrange multiplier ( ) After the global rigidity matrix is obtained by ANSYS, the average equivalent strain of formula (A-19) is obtained by formula (A-26). It can also be directly obtained by ANSYS.

Update process based on morphology:

In order to ensure the connectivity of the topology optimization results, this embodiment adopts the bwlabel function in MATLAB to judge the structural connectivity by four neighbors, and uses morphological operations as a filtering scheme. Combined with the morphological operation function (bwmorph) in MATLAB, a morphological update process suitable for the SIMP method is proposed.

This embodiment uses four subfunctions in the morphological operation number bwmorph of MATLAB, namely bridge, diag, fill and spur. Bridge can connect pixels within one pixel, which helps to connect topological structures; diag can make eight-neighbor blocks become four-neighbor blocks, avoiding single-point connection in the topological optimization structure; fill can fill internal holes, increase structural stability and reduce manufacturing difficulties; spur can reduce redundant branch pixels and reduce useless branch structures.

In detail, this embodiment proposes an update process based on morphology, the flow chart of which is shown in FIG17 , and FIG16 is an actual operation example. The overall process is as follows:

Step 1: At the beginning of the topology optimization process, set the initial value of the binary valve threshold according to different design intervals and boundary conditions.

Step 2: Convert the entity density into a binary image BW according to the binary threshold value. If the entity density is greater than or equal to the threshold , it is 1 in the binary image; if it is less than the threshold , it is 0.

Step 3: Perform morphological operations on the binary image BW. Use the bwmorph function in MATLAB to obtain the binary image through bridge, diag, fill and spur subfunctions. .

Step 4: The bwlabel is used to identify the blocks in a four-neighbor manner, and the redundant blocks that are not connected to the boundary conditions are filtered out. The element set of this redundant block is called .

Step 5: Binarize the image after morphological operation Compare with the original binary image BW to get the total set of newly added elements and delete elements from a collection .

Step 6: Combine the elements of the redundant blocks and remove elements from a collection is the total number of deleted elements .

Step 7: Based on the total number of newly added elements and the total number of deleted elements The design variables and entity density are updated through equations (A-52) and (A-53).

Formula (A-52): ;

in A collection of total newly added elements.

Formula (A-53): ;

in is the total set of deleted elements.

Step 8: Adjust the binarization threshold according to formula (A-54). As the topology optimization results gradually converge, the entity density will tend to be binarized. Therefore, the binarization threshold must be adjusted with iteration. The judgment standard is the volume ratio of the entity density of the current iteration and the volume ratio of the binarized image. This can make the volume ratios before and after the conversion similar to avoid distortion of the binarized image.

Formula (A-54): ;

in is the volume ratio of the binary image of the current generation, and volfrac is the volume ratio of the entity density of the current generation.

7. Topology optimization process of flexible isomechanical mechanism:

The overall process and parameter setting of the design method of the flexible isomechanical mechanism with topology optimization of this embodiment will be described in detail below. For the specific flow chart, please refer to FIG. 18.

The operation process of the flexible isomechanical mechanism topology optimization method is as follows:

Step 1: Define the design interval and the corresponding boundary conditions according to the design requirements, and set the design parameters and initial values. The target output force is F _obj , and the initial value is set to a guess value; the valve value of the parametric projection method Set to 0.5, projection parameters The initial value is 1; the material parameters of the hyperelastic body Defined according to formula (A-18), and given The initial value of the parameter is , equivalent strain valve value At the same time, according to the MMA parameter settings in Table 1, the optimization problem of this embodiment is converted into the MMA square sum minimization form, and the solution parameters of the MMA sub-problem are set according to Table 2.

Step 2: Perform finite element analysis with solid density and defined boundary conditions. Here, refer to Figures 5 to 8 to set virtual springs according to different design intervals, and divide the boundary conditions into virtual models and real models for analysis according to Figures 9 and 10.

Step 3: Obtain the average equivalent strain from the results of finite element analysis , and update the hyperelastic material parameters according to formula (A-19) , which is used in the next iteration of finite element analysis. At the same time, the element sensitivity of the MMA constraint formula is calculated using formula (A-51) based on the finite element analysis results.

Step 4: Use MMA to update the design variables by substituting the design variables of the current inner loop iteration and the element sensitivity of the MMA constraint calculated in step 3.

Step 5: Convert the updated design variables in step 4 into filter variables through the filtering algorithm of formula (A-47).

Step 6: Convert the filtered variable into solid density via the projection function of equation (A-49).

Step 7: Update the design variables and entity density using equations (A-52) and (A-53) through the aforementioned morphological update process.

Step 8: Determine whether the inner loop is converged. The convergence condition of the inner loop must meet the force change less than F _tol defined in (A-14) and the projection parameter. must be equal to 512. If the convergence condition is met, proceed to step 9. If the convergence condition is not met, the number of inner loop iterations loopbeta = loopbeta + 1. If loopbeta reaches 50 times or the force change is less than F _tol , the projection parameter And loopbeta = 0, and return to step 2.

Step 9: Determine whether the outer loop is convergent and the force error Is it less than the allowable value errtol ? If not, correct the target output force F _obj according to formula (A-13) and return to step 1.

In the following, an innovative flexible isotropic mechanism is designed as an example based on the above-mentioned flexible isotropic mechanism topology optimization method.

Regarding boundary conditions:

Please refer to Figures 19 and 20. This embodiment refers to the combination method of the flexible isoforce end effector of Evans and Howell, and multiple flexible isoforce mechanisms (1) are symmetrically mounted on the outer sides of the robot arm end fixture (2) and the tool end fixture (3). One end of the flexible isoforce mechanism (1) is fixed to the robot arm end fixture (2) of the matching robot arm, and the other end is fixed to the tool end fixture (3) of the matching processing tool. Different numbers of flexible isoforce mechanisms (1) can be added according to needs.

According to the combination of components in FIG20, the boundary conditions of the flexible isotropic mechanism of this embodiment are defined. As shown in FIG21, the boundary condition setting can be divided into a virtual model with a virtual spring added to the output end and a real model without a virtual spring, respectively giving a real input displacement and three virtual input displacements at the input end, wherein the range of the three virtual input displacements represents the isotropic interval, and the corresponding virtual output displacement and virtual spring constant are used to represent the virtual output force. and They represent the virtual input displacement and the real input displacement, both of which are upward. and They represent the virtual output displacement and the real output displacement respectively, both of which are upward. k _out represents the virtual spring constant at the output end, and F _out represents the virtual output force. The boundary conditions are expressed by limiting the input and output of the mechanism to displacement in the y direction only, and adding a non-design interval (1 mm × 8 mm) of empty elements between the input and output to avoid generating solid elements that hinder the compression action of the mechanism. The original design interval of the flexible isomechanical mechanism is a 30 mm × 80 mm rectangle, which is discretized with square elements, and TPE is selected as the material of the square elements. The specific parameter settings can be referred to in Table 3.

Table 3 Parameter setting table of flexible isodynamic mechanism: Parameters Numerical value True input displacement (mm) 2 Target actual output displacement LimitL (mm) 2 Virtual input displacement (mm) 2, 3, 4 Target Power (N) 0.45 Virtual spring constant (N/mm) 100 Square element side length (mm) 1 Square element thickness (mm) 1 Young's modulus (MPa) 42.82 Pusombi 0.45 Target volume rate 0.15 Initial value of design variable x _i 0.15 Inner loop convergence error tolerance 0.001 Outer loop convergence error tolerance errtol 0.03 Penalty coefficient p 3 Filter radius r (unit element) 1.5 Equivalent strain valve value 0.5 Binarization threshold initial value 0.25

According to the boundary conditions and design parameters set above, the above-mentioned flexible isomechanical mechanism topology optimization method is used. The optimization process and results are shown in Figure 22, where (a) is the second inner loop iteration, (b) is the 10th inner loop iteration, (c) is the 50th inner loop iteration, (d) is the 150th inner loop iteration, (e) is the 250th inner loop iteration, and (f) is the 352nd inner loop iteration. It can be seen that the result has no unconnected structures and single-point connection parts, and has good connectivity.

Please refer to FIG23. Since the topology optimization result is composed of square elements, the final result has a saw-toothed profile. In order to avoid interference of the edge profile during operation and to facilitate manufacturing, the flexible isotropic mechanism (1) of this embodiment can further smooth the saw-toothed profile by using the commercial software SolidWorks. The smoothed shape is shown in FIG24.

Please refer to FIG. 25 . The design result of the above-mentioned flexible isotropic mechanism (1) is further described below. A fixing portion (111) including a mounting hole can be added to both ends (11) and manufactured by 3D printing. The material can be, for example, a flexible material such as Filastic ^TM TPE from BotFeeder. The filling rate can be set to 100%. The thickness of the flexible isotropic mechanism (1) can be, for example, 5 mm or 10 mm.

The above-mentioned flexible isotropic mechanism is further tested for its actual isotropic effect. The testing method is to fix a tension and pressure gauge on an input slide of a manual slide, and one end of the flexible isotropic mechanism is fixed on a fixed seat of the manual slide, and the other end is fixed on a movable slide between the input slide and the fixed seat. In this way, displacement input can be given through a screw, so that the tension and pressure gauge pushes the movable slide, thereby measuring the force output when the flexible isotropic mechanism is compressed. This experiment is divided into a loading experiment and an unloading experiment. The experimental results can be referred to as shown in Figures 26 and 27, and the effect of equal force output can be clearly observed, wherein Figure 26 is the experimental result of the flexible equal force mechanism with a thickness of 5 mm; Figure 27 is the experimental result of the flexible equal force mechanism with a thickness of 10 mm.

Please refer to Figures 28, 29 and 30. The number of the flexible isoforce mechanisms (1) used can be adjusted according to the application requirements. Please refer to Figure 31. To ensure the stability of the flexible isoforce end effector, the flexible isoforce end effector can include multiple flexible isoforce mechanisms (1) that are opposite to each other, and an optical axis assembly (4) can be installed between the robot arm end fixture (2) and the tool end fixture (3) so that the robot arm end fixture (2) and the tool end fixture (3) can stably move up and down relative to each other. Here, the robot arm end fixture (2) and the tool end fixture (3) are further installed with a force sensor and an electric grinder, respectively, to verify the actual isoforce effect of the flexible isoforce end effector.

Flexible isoforce end effectors can be used in processes such as grinding and polishing, spot welding and gluing. In order to simulate the actual use of flexible isoforce end effectors in the process, a dynamic surface contact test is designed. As shown in Figure 32, it is a cross-sectional diagram of the contact surface (C) to simulate the force condition when the tool end contacts the undulating surface after the flexible isoforce end effector is assembled. The experimental method is to fix the contact surface (C) on the y-axis slide, and the electric grinder is installed on the lower side surface of the flexible isotropic end effector. The initial state of the experiment is to fit the roller probe to the left side of the contact surface (C). After the experiment starts, the flexible isotropic end effector remains fixed as a whole, and the components of the contact surface (C) are driven by the y-axis slide to move relatively, so that the probe scans the contact surface (C) until the probe contacts the end of the surface (C), and the force value of the sensor during this process is recorded. In this experiment, the thickness of the flexible isotropic mechanism (1) is 10 mm.

The experimental results are shown in Figures 33 and 34, where _Fz represents the force in the z direction, _Fy represents the force in the y direction, and the first area (C1) is 24 mm to 64 mm, which is used to observe the actual effect of the flexible isoforce end effector when it is actuated in the isoforce range of 5 mm to 10 mm; the second area (C2) is 64 mm to 94 mm, which is used to observe the actual effect of the flexible isoforce end effector when it is actuated in the isoforce range of 5 mm to 7 mm.

The above end effector test results have an isoforce range of 5 mm to 13 mm, an isoforce range length of 8 mm, an isoforce value of 4.3 N, an average absolute error of 2.55%, a maximum amplitude of 4.30%, and an isoforce range to mechanism length ratio of 8.3%. This experimental data is compared with the existing flexible isoforce mechanism as shown in Table 4.

Table 4 Comparison of elastic isomechanical mechanisms: Project Isotropic interval Mechanism length Equal force amplitude Pham and Wang 7 mm 113 mm 6.2% 18% Wang and Lan 2.5 mm 140 mm 1.8% 10% Wei and Xu 1 mm 38 mm 2.6% 10.5% Chen and Lan 5 mm 100 mm 5% 8% This embodiment 8 mm 96 mm 8.3% 4.3%

The above Pham and Wang data are taken from: H.-T. Pham and D.-A. Wang, "A constant-force bistable mechanism for force regulation and overload protection," Mechanism and Machine Theory, vol. 46, no. 7, pp. 899-909, 2011.

The above data of Wang and Lan are taken from: J.-Y. Wang and C.-C. Lan, "A constant-force compliant gripper for handling objects of various sizes," Journal of Mechanical Design, vol. 136, no. 7, 071008, 2014.

The above Wei and Xu data are taken from: Y. Wei and Q. Xu, "Design of a new passive end-effector based on constant-force mechanism for robotic polishing," Robotics and Computer-Integrated Manufacturing, vol. 74, 102278, 2022 ..

The above data of Chen and Lan are taken from: Y.-H. Chen and C.-C. Lan, "An Adjustable Constant-Force Mechanism for Adaptive End-Effector Operations," Journal of Mechanical Design, vol. 134, no. 3, 031005, 2012.

It should be noted that the above contents are only preferred embodiments of the present invention, and the purpose is to enable persons of ordinary skill in the art to understand the present invention and implement it accordingly, and are not intended to limit the scope of the patent application of the present invention; therefore, all equivalent changes or modifications of the present invention are covered by the scope of the patent application.

(1): Flexible isotropic mechanism (11): End (111): Fixed part (2): Robot arm end fixture (3): Tool end fixture (4): Optical axis assembly (C): Contact surface (C1): First area (C2): Second area

FIG. 1 is a schematic diagram of the schematic flow of the topology optimization method of the embodiment of the present invention. FIG. 2 is a schematic diagram of the design interval of the embodiment of the present invention. FIG. 3 is a schematic diagram of the virtual spring model of the embodiment of the present invention. FIG. 4 is a schematic diagram of the virtual spring model of the embodiment of the present invention. FIG. 5 is a schematic diagram of the boundary condition setting of the virtual spring added to the embodiment of the present invention. FIG. 6 is a schematic diagram of the boundary condition setting of the virtual spring added to the embodiment of the present invention. FIG. 7 is a schematic diagram of the virtual spring fixed at an infinite distance of the embodiment of the present invention. FIG. 8 is a schematic diagram of the virtual spring fixed at an infinite distance of the embodiment of the present invention. FIG. 9 is a schematic diagram of the boundary condition model of the embodiment of the present invention. FIG. 10 is a schematic diagram of the boundary condition model of the embodiment of the present invention. FIG. 11 is a diagram showing the relationship between the input displacement and the output force of the original flexible mechanism. FIG. 12 is a diagram showing the relationship between the input displacement and the output force of the flexible isomechanical mechanism. FIG. 13 is a schematic diagram of the hyperelastic body assumption method. FIG. 14 is a schematic diagram of the filter radius. FIG. 15 is a schematic diagram of the parametric projection method under different projection parameters. FIG. 16 is a schematic diagram of an update example based on morphology. FIG. 17 is a schematic diagram of an update process based on morphology. FIG. 18 is a detailed flow chart of the design method of the flexible isomechanical mechanism with topology optimization of the embodiment of the present invention. FIG. 19 is a three-dimensional schematic diagram of the flexible isotropic end effector component assembly of the embodiment of the present invention. FIG. 20 is a cross-sectional schematic diagram of the flexible isotropic end effector component assembly of the embodiment of the present invention. FIG. 21 is a schematic diagram of the boundary conditions of the flexible isotropic mechanism of the embodiment of the present invention. FIG. 22 is an evolution diagram of the topological optimization results of the flexible isotropic compression mechanism of the embodiment of the present invention. FIG. 23 is a contour diagram of the flexible isotropic mechanism of the embodiment of the present invention before smoothing. FIG. 24 is a contour diagram of the flexible isotropic mechanism of the embodiment of the present invention after smoothing. FIG. 25 is a three-dimensional appearance schematic diagram of the flexible isotropic mechanism of the embodiment of the present invention with a fixed portion added. FIG. 26 is the first experimental data diagram of the isodynamic effect of the flexible isodynamic mechanism of the embodiment of the present invention. FIG. 27 is the second experimental data diagram of the isodynamic effect of the flexible isodynamic mechanism of the embodiment of the present invention. FIG. 28 is a three-dimensional schematic diagram of the flexible isodynamic end effector of the embodiment of the present invention including three flexible isodynamic mechanisms. FIG. 29 is a three-dimensional schematic diagram of the flexible isodynamic end effector of the embodiment of the present invention including six flexible isodynamic mechanisms. FIG. 30 is a three-dimensional schematic diagram of the flexible isodynamic end effector of the embodiment of the present invention including four flexible isodynamic mechanisms. FIG. 31 is a three-dimensional schematic diagram of the appearance of the flexible isodynamic end effector of the embodiment of the present invention with an optical axis assembly installed. FIG. 32 is a schematic cross-sectional view of the contact surface used in the experiment of the embodiment of the present invention. FIG. 33 is the experimental data diagram 1 of the dynamic surface contact test of the embodiment of the present invention. FIG. 34 is the experimental data diagram 2 of the dynamic surface contact test of the embodiment of the present invention.

(1): Elastic isomechanical mechanism

(11): End

(111):Fixed part

Claims (10)

A design method of a flexible isomechanical mechanism of topological optimization design is to design a flexible isomechanical mechanism by executing a topological optimization method on a computer. The flexible isomechanical mechanism corresponds to a design interval and is discretized into a plurality of elements. Each of the elements has a corresponding element density xi _, which is used as a design variable of the topological optimization method. The topological optimization method includes the steps of calculating an optimization problem. The optimization problem includes: an objective function min:

; One-limit formula: R ( U ) = 0;

0< x _min

x _i

1,i=1,2,…, Ne _; Error formula:

;

, pn =2,3,4; where hi is the error formula of the objective function, R is a nonlinear equilibrium residual under a virtual model, U _is a global displacement vector under the virtual model,

is a nonlinear equilibrium residual under a real model,

is _a global displacement vector under the real model, Ne represents the number of elements,

is the element density x _i calculated by a density filtering evolution method and a parameterized projection method, ν _i is the volume of the i-th element in the design interval, V ^* is a target volume, x _min is a preset minimum value of a design variable, h ₁ is the error formula of the true output displacement,

represents the actual output displacement error under the pnth input process. The subscript pn represents the number of different input processes.

represents the actual output displacement, LimitL represents the target value of the actual output displacement,

is the virtual output force error under the pn -th input process, F _out,pn is the output force under the pn -th input process, and F _obj is the target output force; wherein the design interval limits both a mechanism input end and a mechanism output end to displacement only in a y direction. A design method for a flexible isomechanical mechanism with topological optimization design as described in claim 1, wherein the topological optimization method is analyzed by a geometric nonlinear finite element method, and the geometric nonlinear finite element method is added with a hyperelastic body assumption method, and the hyperelastic body assumption method is to add a hyperelastic body material to the region with lower element density in the design range for calculation. As described in claim 1, the design method of a flexible isomechanical mechanism with topological optimization design adopts the method of moving asymptotes (MMA) as a variable updating method, approximately transforms the target function into a separable convex function sub-problem, and then solves it by a dual method. A design method for a flexible isomechanical mechanism with topological optimization design as described in claim 1, wherein the density filtering algorithm calculates the weighted average of the design variables of the surrounding elements contained in a filtering radius of any of the elements, and the parameterized projection method binarizes the design variables. The design method of a flexible isomechanical mechanism of topological optimization design as described in claim 4, wherein an image after the design variable is binarized is updated by a morphological operation algorithm, and the morphological operation algorithm includes a connection step, a filling step, a branch reduction step, and/or an expansion step. The connection step is to connect pixels within one pixel in the image, the filling step is to fill an internal hole in the image, the branch reduction step is to reduce a redundant branch pixel in the image, and the expansion step is to expand the pixels connected by a single point in the image. A design method for a flexible isomechanical mechanism of topological optimization design is to design a flexible isomechanical mechanism by executing a topological optimization method on a computer. The flexible isomechanical mechanism corresponds to a design interval and is discretized into a plurality of elements. Each of the elements has a corresponding element density, which is used as a design variable of the topological optimization method. An image of the design variable after binarization is updated by a morphological operation algorithm to ensure the connectivity of a topological optimization result of the image. The design interval limits a mechanism input end and a mechanism output end to displacement in a y direction only. A topology-optimized flexible isomechanical mechanism is manufactured using a design method for a topology-optimized flexible isomechanical mechanism as described in any one of items 1 to 6 of the patent application scope, using flexible materials, and the topology-optimized flexible isomechanical mechanism includes two opposite ends for connecting a robot arm end fixture and a tool end fixture, respectively. A topologically optimized flexible isotropic end effector includes a plurality of topologically optimized flexible isotropic mechanisms as described in Item 7 of the patent application, a robot end fixture and a tool end fixture, the two ends of which are respectively connected to the robot end fixture and the tool end fixture, the tool end fixture is used for grinding, polishing, spot welding or gluing processes, and an optical axis assembly is installed between the robot end fixture and the tool end fixture. A computer program product stores a program, and when the program is loaded and executed on a computer, the design method of a flexible isotropic mechanism with topology optimization design as described in any one of claim items 1 to 6 can be completed. A computer-readable recording medium storing a program, when the computer loads and executes the program, can complete the design method of a flexible isomechanical mechanism with topology optimization design as described in any one of claim items 1 to 6.

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