TWI795114B - Multimaterial topology optimization method of adaptive compliant actuator, grippers , method, computer program product and computer readable recording medium for designing such - Google Patents

Abstract

The invention provides a multimaterial topology optimization method for design of adaptive compliant actuators and grippers. The multimaterial topology optimization method is based on solid isotropic material with penalization. The multimaterial topology optimization method can improve the objective function by increasing the flexibility of the use of materials.

Description

Adaptive flexible fingers, grippers, design methods, computer program products, and computer-readable recording media for multi-material condition design

The invention relates to a clamping tool, especially an adaptive flexible finger designed with multi-material conditions, a gripper, a design method, a computer program product, and a computer-readable recording medium.

Flexible grippers are suitable for objects with large differences in shape, and can reduce the chance of clamping damaged objects, so they are widely used. Related cases include, for example, "Flexible Grippers and Design Methods, Computer Program Products, and Computer-Readable Recording Media" of the Republic of China Invention Patent Announcement No. I630499 approved by the inventor.

The flexible grippers disclosed in the above patents are integrally formed of elastic materials, but different parts of the flexible grippers have different rigidity requirements. Deformation to better wrap the object.

；步驟三：執行有限元素分析，以求得節點位移，其中元素實體密度

與楊氏係數E _i 間之關係為：

其中

、

其中

代表第一元素實體密度，上標p為懲罰係數，E _min為趨近於零之極小值，M值為所使用之材料數量再加上1，Π為連乘運算子；步驟四：以節點位移計算目標函數及元素靈敏度；步驟五：更新各種材料之元素密度；步驟六：判斷是否收斂，若收斂即結束；否則至步驟二進行下一次疊代。 Therefore, in order to improve the practicability of adaptive flexible fingers, a design method of adaptive flexible fingers based on multi-material condition design is proposed to perform topology optimization through a computer-based SIMP method (Solid isotropic material with penalization method) , including the following steps: Step 1: Obtain information about the design interval, boundary conditions and parameters related to topology optimization, and obtain the initial value of the element density ρ _i of a plurality of materials, the subscript i represents the i- th element in the design interval Element; Step 2: filter the element density ρ _i of various materials respectively, so as to convert the element density ρ _i of various materials into the element density

;Step 3: Perform finite element analysis to obtain the node displacement, where the element solid density

The relationship with Young's coefficient E _i is:

in

,

in

Represents the physical density of the first element, the superscript p is the penalty coefficient, E _min is the minimum value close to zero, M is the amount of materials used plus 1, and Π is the multiplication operator; step 4: use the node Displacement calculation objective function and element sensitivity; step 5: update the element density of various materials; step 6: judge whether it is converged, if it converges, it will end; otherwise, go to step 2 for the next iteration.

；其中v _i 為元素之體積，V為設計區間之體積，V ₀ ^*則為目標體積率。 Further, the volume restriction formula in the SIMP method is:

; Where v _i is the volume of the element, V is the volume of the design interval, and V ₀ ^* is the target volume ratio.

Further, the topology optimization method uses an element density filtering algorithm and a critical projection method to update the design variables in sequence. In addition, the topology optimization method can add robust topology optimization to simultaneously minimize the objective functions of the original design, the expanded design and the reduced design.

Further, the SIMP method may use mutual potential energy (Mutual potential energy, MPE ) as an objective function, and use a suitable optimized objective function for each knuckle in the adaptive flexible finger.

With the above features, the following effects can be mainly achieved:

1. The topology optimization method of the present invention can improve self-adaptability through multi-material condition design.

2. The topology optimization method of the present invention can further optimize the performance of the design in different operating scenarios by increasing the number of boundary conditions.

3. The variable update of the present invention can further add filtering algorithm and critical projection method, the former can be used to solve the problem of checkerboard and mesh dependency, and the latter is used to solve the problem of gray scale elements.

4. The topology optimization method can further add robustness topology optimization, and perform topology optimization on the original design, expansion design and reduction design at the same time, so as to avoid the problem of unreasonable hub structure in the design (such as manufacturing Difficulty on the surface, prone to fatigue or stress concentration and other poor durability issues).

(1): Adaptive flexible fingers

(10): terminal knuckles

(20)(30): middle knuckles

(40):first knuckle

(50): Displacement transfer module

(51): Displacement transfer structure

(52): Actuator

(521):Moving parts

(522):Fixer

(523): actuator

(60A)(60B): hub structure

(100): clamping surface

(S01): Step 1

(S02): Step 2

(S03): Step 3

(S04): Step 4

(S05): Step five

(S06): Step 6

(M01): The first flexible material

(M02): Second flexible material

Fig. 1 is a flow chart of topology optimization according to an embodiment of the present invention.

FIG. 2A is a schematic diagram of a design interval according to an embodiment of the present invention.

Fig. 2B is a second schematic diagram of the design interval of the embodiment of the present invention.

Fig. 3A is a schematic diagram of the general and solid design intervals of the embodiment of the present invention.

Fig. 3B is a second schematic diagram of the general and solid design intervals of the embodiment of the present invention.

Figure 3C is a schematic diagram of the original design interval.

Figure 3D is a schematic diagram of the results of multi-material design in the original design interval.

Figure 3E is the second schematic diagram of the multi-material design results of the original design interval.

Figure 4 is a schematic diagram of the degrees of freedom of element nodes.

Figure 5A is a schematic diagram of a checkerboard grid.

Fig. 5B is a schematic diagram of the result without adding the filtering algorithm and the critical projection method.

Fig. 5C is a schematic diagram of the result of adding the filtering algorithm.

Fig. 5D is a schematic diagram of the result of adding filtering algorithm and critical projection method.

Figure 6 is a schematic diagram of the filtration radius.

Fig. 7 is a schematic diagram of the projection function.

Fig. 8A is the first schematic diagram of the design result of the projection function.

Fig. 8B is the second schematic diagram of the design result of the projection function.

Fig. 8C is the third schematic diagram of the design result of the projection function.

Fig. 8D is a schematic diagram 4 of the design result of the projection function.

Fig. 9A is the projection function after adjusting the threshold value (η=0.3).

Fig. 9B is the projection function after adjusting the threshold value (η=0.7).

Fig. 10 is a flow chart of convergence analysis of the embodiment of the present invention.

Fig. 11 is a schematic diagram of the design interval of the flexible mechanism of the embodiment of the present invention.

Fig. 12A is a first schematic diagram of the superposition principle of the embodiment of the present invention.

Fig. 12B is the second schematic diagram of the superposition principle of the embodiment of the present invention.

Fig. 13 is a schematic diagram of a design interval of a single input and a double output according to an embodiment of the present invention.

Fig. 14A is a first schematic diagram of dismantling boundary conditions of single input and dual output according to the embodiment of the present invention.

Fig. 14B is a second schematic diagram of dismantling boundary conditions of single input and dual output according to the embodiment of the present invention.

Fig. 15 is a schematic diagram of the design interval of dual-input and dual-output according to the embodiment of the present invention.

Fig. 16A is a first schematic diagram of dismantling boundary conditions of dual-input and dual-output according to the embodiment of the present invention.

Fig. 16B is a second schematic diagram of dismantling boundary conditions of dual-input and dual-output according to the embodiment of the present invention.

Fig. 17 is a detailed flowchart of topology optimization according to the embodiment of the present invention.

FIG. 18 is a flow chart of code judgment conditions according to an embodiment of the present invention.

Fig. 19 is a schematic diagram of the structure of the self-adaptive flexible finger according to the embodiment of the present invention.

Fig. 20 is a schematic diagram of the gripping strategy of the self-adaptive flexible finger according to the embodiment of the present invention.

Fig. 21A is a first schematic diagram of the boundary conditions of the self-adaptive flexible finger according to the embodiment of the present invention.

Fig. 21B is the second schematic diagram of the boundary conditions of the self-adaptive flexible finger according to the embodiment of the present invention.

Fig. 21C is the third schematic diagram of the boundary conditions of the self-adaptive flexible finger according to the embodiment of the present invention.

Fig. 21D is a fourth schematic diagram of the boundary conditions of the self-adaptive flexible finger according to the embodiment of the present invention.

Fig. 22 is a schematic diagram of the boundary conditions of the displacement transfer structure of the embodiment of the present invention.

Fig. 23A is a first schematic diagram of the multiple boundary conditions of the phalanx in the embodiment of the present invention.

Fig. 23B is a second schematic diagram of the multiple boundary conditions of the phalanx in the embodiment of the present invention.

Fig. 24 is a schematic perspective view of the three-dimensional appearance of the self-adaptive flexible finger of the embodiment of the present invention.

Fig. 25 is a three-dimensional appearance schematic diagram of the clamping jaw of the embodiment of the present invention.

Fig. 26 is the second perspective view of the clamping jaw of the embodiment of the present invention.

Fig. 27 is a three-dimensional appearance schematic diagram of the clamping jaw of the embodiment of the present invention.

Fig. 28 is a schematic view four of the three-dimensional appearance of the clamping jaw of the embodiment of the present invention.

Based on the above-mentioned technical features, the main functions of the self-adaptive flexible fingers, grippers, design methods, computer program products, and computer-readable recording media designed with multi-material conditions of the present invention can be clearly presented in the following embodiments with diagrams. It should be noted that, for ease of understanding, in each drawing, similar functional components will use similar or identical component symbols.

The self-adaptive flexible finger design method of multi-material conditional design in the embodiment of the present invention can be implemented as a program and stored in a computer program product or a computer-readable recording medium. After the computer loads and executes the program, the adaptive flexible finger design method as described above for multi-material condition design can be completed.

The adaptive flexible finger design method of multi-material condition design includes designing with a topology optimization method, and the topology optimization is mainly based on the SIMP method (Solid Isotropic Material with Penalization). It has better calculation speed and no mesh dependency, and is suitable for flexible mechanism design.

The following will introduce the process and theory of 3D topology optimization in sequence, including design interval, design variables, finite element analysis, filtering algorithm, robust topology optimization, MMA method, convergence criterion, objective function and Element sensitivity.

Please refer to Figure 1 first, which reveals the main flow of the topology optimization method described in this embodiment. It is based on the SIMP method, and performs filtering algorithms and critical projections on design variables (ie, element density). The method can avoid the structure to produce a checkerboard grid and reduce the number of grayscale elements. Therefore, the final design variable will be close to 0 or 1. The process flow of the topology optimization method is as follows:

Step 1 (S01): Define the design interval, boundary conditions, and parameters related to topology optimization, and obtain the initial value of the element density (also called the initial value of the design variable).

Step 2 (S02): Perform a filtering algorithm and a critical projection method on the element density.

Step 3 (S03): Perform finite element analysis and calculate the displacement of nodes.

Step 4 (S04): Calculate the objective function and element sensitivity based on node displacement

Step 5 (S05): Updating design variables by MMA method.

Step 5 (S05): Judging whether it is converged, if it is converged, it will end; otherwise, go to step 2 (S02) for the next iteration.

The design interval, design variables and finite element analysis will be further described in detail below: the self-adaptive flexible gripper of this embodiment adopts the topology optimization method of the two-dimensional design interval. Taking the two-dimensional cantilever beam as an example, Figure 2A shows the cantilever beam subjected to plane stress. After the relevant dimensions, external forces and boundary conditions are completely defined, the design interval in topology optimization can be divided . As shown in Figure 2B, the element quantity in the x direction in the design interval is called nelx, and the element quantity in the y direction is called nely. It can be observed from Figure 2B that the nelx of the design interval of the cantilever beam is 12, the nely is 6, and the design interval has been divided into 72 square elements.

The design interval of the flexible gripper in this embodiment can be further divided into general design interval and solid design interval, as shown in Figure 3A and Figure 3B, the elements in the solid design interval must remain solid elements during the topology optimization process . However, some elements in the solid design interval may still be cut out during the variable update process, so the solid elements in the solid design interval will be maintained by forcing the element back to a solid element.

，能夠代表實際問題模型的機械性質，例如剛性。而元素實體密度為將元素密度經由執行濾化演算法及臨界投影法所求得。在SIMP方法中，元素實體密度與楊氏係數(E _i )間之關係如式(A-1)所示，並且在有限元素法中的元素剛性矩陣(k _i )為楊氏係數呈線性關係，如式(A-2)所示：

In the topology optimization method, each element has an independent element density ( ρ _i ), its value is between 0 and 1, and the subscript i represents the i- th element in the design interval, which is the topology Design variables in Pak optimization. The density of each element determines whether the element will be retained or removed. When it is 0, it means that the element is an empty element and will be removed. When it is 1, it means that the element will be retained and is a solid element. Elemental Entity Density

, which can represent the mechanical properties of the actual problem model, such as stiffness. The element entity density is obtained by performing the filtering algorithm and the critical projection method on the element density. In the SIMP method, the relationship between the element solid density and Young's coefficient ( E _i ) is shown in formula (A-1), and the element stiffness matrix ( k _i ) in the finite element method has a linear relationship with Young's coefficient , as shown in formula (A-2):

k _i = E _i k ₀ (A-2)

為1時，E _i 等於E ₀，表示該元素為實心元素，其楊氏係數與原先所設定之拓樸材料相同。當

為0時，E _i 等於E _min ，表示該元素為空元素，其楊氏係數為趨近於零之正數，此做法是要避免在有限元素分析時產生奇異(Singularity)的全域剛性矩陣。在拓樸最佳化過程中的結構呈現上，實心元素呈現黑色元素，空元素則呈現為白色元素。而當

介於0與1之間時，則稱為灰階元素，在呈現上為灰色。此種元素雖然有助於在疊代過程中找出最佳解，但會導致最終設計結果的模糊化。因此加入懲罰係數的目的即是希望能將灰階元素趨向實心元素或空元素。 Equation (A-1) is the formula for defining the Young's modulus corresponding to the solid density of each element. E ₀ is the Young's modulus of the original material, and E _min is the minimum value close to zero, which can be set to 10 ^-9 , p is the penalty coefficient, which can be set to 3, and k ₀ in the formula (A-2) is the element rigidity matrix of the material whose Young's modulus is 1. It can be observed from formula (A-1) that when

When it is 1, E _i is equal to E ₀ , which means that the element is a solid element, and its Young's modulus is the same as that of the previously set topological material. when

When it is 0, E _i is equal to E _min , indicating that the element is an empty element, and its Young's coefficient is a positive number close to zero. This method is to avoid a global rigid matrix of singularity during finite element analysis. In the structure presentation during the topology optimization process, the solid elements appear as black elements, and the empty elements appear as white elements. And when

When it is between 0 and 1, it is called a grayscale element, and it is gray in presentation. Although such elements help to find the best solution in the iterative process, they can lead to blurring of the final design results. Therefore, the purpose of adding the penalty coefficient is to make the grayscale elements tend to be solid elements or empty elements.

、

及

分別為第一、第二及第三元素實體密度。第一元素實體密度決定該元素為實心元素或空元素，而第二及第三元素則進一步決定該元素將使用何種材料。依據式(A-6)，若

，E _i 將帶入E _min。若

，

，E _i 將帶入E ₃。若

，

，

，E _i 將帶入E ₂。若

，

，

，E _i 將帶入E ₁，如三材料設計變數與楊氏係數關係表所示。 Equation (A-1) is the SIMP interpolation model of single material. If it is necessary to add multi-material design, Equation (A-1) must be rewritten into Equation (A-3) to Equation (A-5). Where M is the number of materials used plus one. Taking the three-material design used in this embodiment as an example, M is 4, which is shown in formula (A-6) after unfolding. E ₁ , E ₂ and E ₃ are Young's modulus of material 1, material 2 and material 3 respectively, and must satisfy E ₁ > E ₂ > E ₃ . and

,

and

are the first, second and third element solid densities, respectively. The first element's solid density determines whether the element is solid or empty, while the second and third elements further determine what material the element will use. According to formula (A-6), if

, E _i will be brought into E _min . like

,

, E _i will be brought into E ₃ . like

,

,

, E _i will be brought into E ₂ . like

,

,

, E _i will be brought into E ₁ , as shown in the relationship table between three material design variables and Young's coefficient.

Three material design variables and Young's coefficient relationship table:

In addition, although the element densities in the design interval are independent of each other, the number of solid elements still needs to meet the volume constraint formula in the topology optimization problem equation. The volume constraint formula of single material design is shown in formula (A-7) Show. The volume restriction formula of multi-material design is shown in formula (A-8).

則為目標體積率。在多材料設計中，

為決定所有元素為實心元素或空元素之變數，因此式(A-8)即可完整定義出最終設計之目標體積率。本實施例之多材料設計對於個別材料之使用量則無額外限制，此設定之優點為無須事先測試每種材料之最佳化效果，可將所有材料一併納入多材料設計中由拓樸最佳化方法自由選用材料以提升目標函數，因此在多材料設計的設計結果中之材料仍可全部使用同一種材料，如圖3C、圖3D及圖3E所示。 Among them, v _i is the volume of the element, V is the volume of the design interval,

is the target volume ratio. In multi-material design,

In order to determine whether all elements are solid elements or empty elements, the formula (A-8) can completely define the target volume ratio of the final design. The multi-material design of this embodiment has no additional restrictions on the amount of individual materials used. The advantage of this setting is that it is not necessary to test the optimization effect of each material in advance, and all materials can be included in the multi-material design. The optimization method freely selects materials to improve the objective function, so the materials in the design results of multi-material design can still all use the same material, as shown in Figure 3C, Figure 3D and Figure 3E.

In performing the finite element analysis, this embodiment adopts the static analysis of the linear assumption of the two-dimensional plane. The elements of the design interval introduced earlier are the elements in the finite element analysis. There are four nodes around each element, and each node has degrees of freedom in the x and y directions, as shown in Figure 4. After completing the setting of the design interval and adding the boundary conditions of the problem model (such as fixed end and input force, etc.), the finite element balance formula can be obtained as shown in formula (A-9).

KU = F (A-9)

Among them, K is the global rigidity matrix, U is the global displacement vector, and F is the global force vector. Then, the objective function ( f ) and element sensitivity (α _i ) can be obtained by solving the global displacement vector. The objective function and element sensitivity will be described in detail later.

，臨界投影法再以

進行計算形成

，如下表所示。 The filtering algorithm and the critical projection method will be further described in detail below: the filtering algorithm is used to solve the checkerboard and mesh dependency problems, and the critical projection method is used to solve the gray scale element problem. Since the SIMP method discretizes the continuum into independent grids, that is, the element density, it is easy to produce a checkerboard grid design result as shown in Figure 5A. This design result is not only unmanufacturable, but also not a reasonable design result. The grid dependency problem is that the design results change with the different elements divided by the design interval. Fig. 5B, Fig. 5C, and Fig. 5D are design cases before and after adding filtering algorithm and critical projection method. The process to solve the above problem is to perform a filtering algorithm for element density in each iteration and then perform a critical projection method. The filtering algorithm performs filtering on the element density ρ _i to form

, the critical projection method and then

perform calculations to form

, as shown in the table below.

Variable relationship table:

The basic concept of the filtering algorithm is to average the density of each element with the density of surrounding elements, thereby improving the correlation between elements. The formulas are shown in formulas (A-10) and (A-11).

w _ij = max (0 ,r _min - r _ij ) (A-11)

作為代表。 w _ij is the weight between the i -th element and the j -th element, v _j is the area of the j- th element, if the area of each element is the same, it can be regarded as 1, r _min is the filtering radius, r _ij is the distance between the centroid positions of the i- th element and the j -th element. The filtering algorithm will filter each element in the design interval separately. First, use the filtering radius ( r _min ) to draw a circle with the centroid position of the i- th element as the center, and use the centroid positions of the surrounding elements to distinguish Whether it is in the circle, the elements in the circle will be weighted average with the i- th element, as shown in Figure 6. N _e,i is the number of elements within the circle drawn with the centroid position of the i- th element, and the subscript j is the j -th element in the set of N _e,i , and the calculation method of w _ij is r _min The distance between the centroid positions of two elements is deducted, so it can be observed from formula (A-11) that the closer the distance between elements, the greater the influence on each other, so that each element can be related to adjacent elements. The element density after performing the filtering algorithm is represented by the symbol

As a representative.

After the filtering algorithm is completed for each element in the design interval, the critical projection method will be performed next. In the iterative process of the SIMP method, gray-scale elements are produced, which makes the appearance of the design result blurred. The critical projection method is to reduce the number of grayscale elements through the projection function. The formula is as follows:

In the projection function, the change of the element density is mainly manipulated by the projection parameter ( β ) and the threshold value (η). When the threshold value is set to 0.5, it means that the element density of elements with element density greater than 0.5 will increase towards 1 after the critical projection method is executed. On the contrary, if it is less than 0.5, it will decrease towards 0. The speed of increment and decrement depends on the size of β . It can be observed from Figure 7 that when β is 1, the projection function presents an oblique straight line, that is, the element density is not affected by the projection function. As β increases, the projection function will gradually tend to a step function, but if β is set too large at the beginning of the topology optimization process, it will cause drastic changes in the element density during iterations, which will easily cause the results to fail to converge and affect the optimal solution search. Here, β can be set as 1 in the initial setting of topology optimization, and as long as the design result tends to be flat or after 50 iterations, β can be doubled to a maximum of 512. 8A to 8D show the results of different β values in the iterative process.

、

及

)之計算流程為將元素密度依序帶入式(A-13)與式(A-14)中。 In the three-material design, the three element densities have to go through the filtering algorithm and the critical projection method to become the element entity density with physical meaning. Three element solid densities (

,

and

) calculation process is to bring the element density into formula (A-13) and formula (A-14) in sequence.

The element sensitivity (α _i ) is defined as the partial differentiation of the objective function with respect to the element density. The chain rule used in the calculation process is as follows:

And by partial differentiation of formula (A-12), we can get:

Partial differentiation can be obtained from formula (A-10):

Among them, N _e,j is a set of elements that draw a circle at the centroid position of the jth element with a filtering radius and are located within the circle, and k represents the kth element in the set.

In the three-material design, in order to obtain the sensitivity of the first, second and third elements, the objective function must be partially differentiated with respect to the densities of the first, second and third elements, respectively. The calculation of three element sensitivities (α ₁ , α ₂ and α ₃ ) is as follows:

Then by partial differentiation of formula (A-12) and formula (A-10), we can get:

It should be noted that although the above-mentioned embodiment takes the elemental sensitivity of three materials as an example, it is also possible to choose to bring in the elemental sensitivity of different materials and multiple materials. The general formula for multi-material element sensitivity is as follows (A-20-2):

Among them, m is the number of materials, and E _i is substituted by formula (A-3), so that the sensitivities of m elements of m materials can be calculated as α _{m i} , α _{(m-1) i} , α _{(m-2 ) i} ..., α _{1 i} .

The robustness topology optimization will be further explained in detail below: in the process of topology optimization, abnormal hinge structures often appear, and the reason for the abnormal hinge structure is that it can help increase the flexibility of the place to facilitate the rotation of the mechanism and bending operations. However, such a structure will cause difficulties in manufacturing, and it is also easy to cause fatigue or stress concentration in the flexible mechanism, resulting in a decrease in durability. Therefore, the abnormal hinge structure is a structure that should be avoided in the topology optimization process.

Here, the so-called robust topology optimization is to optimize the topology of the original design, expansion design and reduction design at the same time to avoid the occurrence of abnormal hub structures.

求得，而元素實體密度又是由元素密度ρ經濾化演算法與臨界投影法求得，故目標函數寫作f(

(ρ))。限制式的部分除了須滿足有限元素平衡式之外，還須滿足目標體積率限制式以及元素密度範圍之限制式。 Equation (A-21) is the problem equation before adding robust topology optimization. As mentioned above, the objective function f is determined by the element entity density

and the element entity density is obtained from the element density ρ by filtering algorithm and critical projection method, so the objective function is written as f (

( ρ )). In addition to satisfying the finite element balance formula, the part of the restriction formula must also satisfy the target volume ratio restriction formula and the restriction formula of the element density range.

After adding robust topological optimization, the problem equation must be rewritten as equation (A-22). The concept is to simultaneously minimize the objective functions of the original design, the expanded design and the reduced design, that is, to minimize the maximum value (min-max). The problem of minimizing the maximum value can be optimized for the worst objective function, so that each of the designs will be optimized, so that the design results are less prone to abnormal hinge structures.

與

分別代表擴張設計與縮減設計之元素實體密度，當中

與

是經由ρ並調整投影函數中的門檻閥值η計算而得。於此，可將擴張設計與縮減設計之門檻閥值分別設定為0.3與0.7，調整門檻閥值後的投影函數分別如圖9A及圖9B所示。 The superscripts d and e represent the expanded design and reduced design, respectively.

and

Represent the element entity density of the expanded design and the reduced design, respectively, where

and

It is calculated by ρ and adjusting the threshold value η in the projection function. Here, the threshold values of the expanded design and the reduced design can be set to 0.3 and 0.7 respectively, and the projection functions after adjusting the threshold values are shown in Fig. 9A and Fig. 9B respectively.

The problem equation after adding multi-material design is shown in formula (A-23-1):

Among them, the superscript and subscript d , o and e represent the expanded design, the original design and the reduced design respectively, K is the global rigidity matrix, U is the global displacement vector, and F is the global force vector.

決定該元素為實心元素或空元素，因此須符合第四條限制式，即體積限制式。且三種元素密度均需界於0與1之間。 In this embodiment, the problem equation after adding the three-material design is shown in formula (A-23-2). In the three-material design, each element must be composed of ρ ₁ j , ρ ₂ j and ρ ₃ j In order to define the material type of the element, it represents the first, second and third element densities respectively, and the densities of the three elements must be between 0 and 1. Therefore, the objective function and finite element balance must be rewritten. Also, since the first element solid density

Determine whether the element is a solid element or an empty element, so it must meet the fourth restriction, that is, the volume restriction. And the densities of the three elements need to be between 0 and 1.

The MMA theory will be further described in detail below: The MMA method is a non-linear programming method suitable for structural optimization, which can effectively deal with complex problem equations. When using it, the objective function of the original problem must be converted into a sub-problem in the form of a convex function, and the sub-problem should be solved by the dual method. The general formula of the problem equation of the MMA method is as follows:

與

為設計變數x _j 的下界與上界，n為設計變數之數量，y _i 及z為僅存在於MMA問題中的人工變數(artificial variable)，其作用為增加將原始問題方程式轉換為MMA問題時之靈活度。a ₀、a _i 、c _i 及d _i 同樣為僅存在於MMA問題中之常數，需在求解前即設定完成，當中須滿足a ₀>0、a _i

0、c _i

0、d _i

0及c _i +d _i >0。而當a _i >0時須符合a _i c _i >a ₀。 In formula (A-24), f ₀ is the objective function of the original problem equation, f _i is the constraint function in the original problem equation, m is the number of constraints, and both f ₀ and f _i must be continuous and controllable design variables The function of partial differentiation, x _j is the design variable in the original problem equation,

and

is the lower bound and upper bound of the design variable x _j , n is the number of design variables, y _i and z are artificial variables that only exist in the MMA problem, and their function is to increase the time when the original problem equation is transformed into an MMA problem of flexibility. a ₀ , a _i , ci and d _i are also constants that only exist in MMA problems, and they need to be set before solving the problem, and a _{0 >} 0, a _i must be satisfied

0, c _i

0. d _i

0 and c _i + d _i >0. And when a _i >0 must meet a _i c _i > a ₀ .

The solution process of the MMA method is as follows:

Step 1: Set the MMA related parameters, the initial value x ⁽⁰⁾ of the design variable, and set the number of iterations to iter =0.

。 Step 2: Calculating f _i (x ^{( iter )} ) and ▽ f _i (x ^{( iter )} ), in order to carry out the approximate transformation of the convex function of f _i to obtain

.

建立MMA子問題，再以對偶法求解。 Step 3: Take

Establish the MMA subproblem, and then solve it by the dual method.

Step 4: Bring the solution of the subproblem into x ^{( iter )} , and return to Step 2 for the next iteration: iter = iter +1.

The subproblems of the MMA method are as follows:

(x)之計算如下：

Approximate function in formula (A-25)

(x) is calculated as follows:

、

及

之計算如下：

among

,

and

The calculation is as follows:

in,

及

為設計變數的上下界，其計算方式如下：

In formula (A-25)

and

is the upper and lower bounds of the design variable, and its calculation method is as follows:

與

分別為下漸近線(Lower asymptote)與上漸近線(Upper asymptote)，且當iter=1及iter=2時：

among

and

They are Lower asymptote and Upper asymptote respectively, and when iter =1 and iter =2:

Here, asyinit is set to 0.5 divided by β in the projection function, the purpose is to avoid the change of β in the iterative process and make the design deviate from the original optimization trend.

3時：

And when iter

3 o'clock:

則為移動漸進線的調整參數，其定義如下：

is the adjustment parameter of the moving asymptotic line, which is defined as follows:

In order to apply the robust topological optimization introduced above, the formula (A-24) is transformed into a min-max type problem equation, and the problem of this type will write the objective functions of the original, expanded and reduced designs into In the restriction formula and make it smaller than the artificial variable z, and then set the artificial variable z as the objective function, the rewritten problem equation is as follows:

Among them, h _i (x) is the objective function calculated by the original, expanded and reduced designs, so P is 3, n is the number of design variables, and g _i (x) represents the volume restriction formula. In order to convert the general formula of MMA into formula (A-39), the parameters need to be set as m = P + Q , f ₀ (x) =0, f _i (x) = h _i (x) and f _{P + i} (x) = g _i (x) , the settings of other coefficients are shown in the table below.

MMA related parameter setting table:

The convergence criterion will be further described in detail below: the last step in the topology optimization method is to judge whether the design result has reached convergence, and the execution time of the convergence judgment in this method is after the MMA variable update. The criteria adopted are whether the target volume ratio is within the tolerance range and whether the update of design variables tends to be smooth. The former is judged by the volume ratio tolerance error voltol , while the latter is judged by the element density change rate change , which is defined as follows: change = max (｜ ρ ^new - ρ ｜) (A-40)

則是由ρ ^new 分別經過濾化演算法與臨界投影法計算所求得的元素實體密度。於此，達到收斂條件須同時通過三項標準，分別是投影函數中的β已提升至512、change需小於0.01以及voltol需小於0.001。其流程如圖10A中虛線方框所示。 Among them, ρ ^new is the element density obtained after the MMA variable is updated, and ρ is the element density before the MMA variable is updated. and

It is the element entity density calculated by ρ ^new through filtering algorithm and critical projection method respectively. Here, to achieve the convergence condition, three criteria must be passed at the same time, namely, the β in the projection function has been increased to 512, the change must be less than 0.01, and the voltol must be less than 0.001. The process is shown in the dotted box in Fig. 10A.

The convergence analysis process of multi-material design is similar to the convergence analysis process of single-material design, but since there is no additional restriction on the volume ratio of individual materials in multi-material design in this study, in order to ensure the update of each material in multi-material design Convergence is achieved, so in the three-material design of this study, change 2 and change 3 are defined separately, which correspond to the calculation of the change rate of the density of the second and third elements respectively, while the original change is for the first element in the multi-material design The formula for calculating the rate of change of density is as follows:

The convergence analysis process of multi-material design is shown in the dotted box in Fig. 10B.

The objective function will be described in further detail below: Interaction Potential Energy

As shown in Figure 11, the design interval for designing a flexible mechanism has an input end and an output end, and the lower end is a fixed end. The input end and output end are respectively subjected to a unit virtual force f _in and f _out , and according to The principle of mechanics can be divided into the force f _in,x and f _out,x of the x component, and the force f _in,y and f _{out,y of the y} component. And a spring k _in and k _out with unequal rigidity are connected to the input end and the output end respectively. The springs at the two places can also be divided into springs k _in,x and k _out,x representing the x component, and springs representing the y component The setting of spring k _in,y , k _out,y , and spring constant will be described later.

Here, the mutual potential energy (MPE) is used as the objective function. Interaction potential energy is used to describe the flexibility of a mechanism when it is subjected to two forces. The spring constants of the two acting forces can be set through different ratios to produce different optimization effects. If k _in is larger than k _out , the flexible mechanism is more likely to generate deformation at the output end and transmit displacement. In this method, when designing the flexible gripper, it is expected that the output terminal can produce a corresponding output displacement when the input terminal is given an input displacement, so the maximum interaction potential energy can be used to describe this goal. Conversely, if k _out is larger than kin , the displacement suffered by the flexible mechanism at the input end is more likely to be transmitted to _the output end in the form of force. To calculate the interaction potential energy, the design interval in Figure 11 must first be split into the design interval with only input load and only output load, as shown in Figure 12A and Figure 12B. )Calculation.

When calculating the interaction potential energy, finite element analysis is used to calculate the boundary conditions with only input load and only output load. Among them, U ₁ and U ₂ represent the global displacement vectors of the above two boundary conditions respectively, F ₁ and F ₂ represent the global force vectors of the above two boundary conditions respectively, and K represents the global rigidity matrix. Then according to Hooke's law (Hooke's law), two sets of finite element balance equations can be obtained:

The global displacement vectors U ₁ and U ₂ can be obtained by solving two sets of finite element balance equations, and then the calculation of the interaction potential energy can be completed according to formula (A-46).

The sensitivity analysis will be further described in detail below: element sensitivity is an important index when updating design variables, and is obtained by partial differentiation of element density by the objective function. However, the global rigidity matrix K is composed of the element entity density, so it is necessary to perform partial differentiation on the element entity density by the objective function, and the calculation method is as follows:

，式(A-47)可整理為：

And because K is a symmetric matrix, so

, Formula (A-47) can be organized as:

The calculation of the partial differential of the element entity density by the global displacement vector in the factor (A-48) is very complicated, so in order to avoid this calculation, the following method will be used for substitution. Firstly, the partial differential of formula (A-45) to the element entity density can get the result of formula (A-49). The global force vector in formula (A-49) is a constant vector, so the result of partial differential is 0.

Formula (A-49) can be obtained after transposition:

Then substitute formula (A-50) into formula (A-48) to get:

The form of a single element in the design interval can be written as:

Among them, u ₁ and u ₂ are the displacement vectors of the boundary conditions of only the input load and only the output load respectively, and the partial differential of the element rigidity matrix to the element solid density can be obtained through formula (A-1) and formula (A-2) have to:

Bring formula (A-53) into formula (A-52) to get:

Finally, the element sensitivity calculation can be completed by substituting formula (A-54) into formula (A-15):

Element sensitivity analysis of multi-material design: In this embodiment, three-material design is adopted, so the objective function needs to perform partial differentiation on the three element densities in order to obtain the element sensitivities (α ₁ , α ₂ and α ₃ ) of the three element densities. The elemental sensitivity of the first elemental density is derived as follows:

Among them, the partial differential of the rigid matrix of the element to the solid density of the element can be obtained through formula (A-2) and formula (A-6):

Substituting Equation (A-57) into Equation (A-56) can obtain the partial differential of interaction potential energy to element entity density:

Finally, substituting formula (A-58) into formula (A-15) is the element sensitivity of the first element density:

The elemental sensitivity of the second elemental density is derived as follows:

Among them, the partial differential of the rigid matrix of the element to the solid density of the element can be obtained through formula (A-2) and formula (A-6):

Substituting Equation (A-61) into Equation (A-60) can obtain the partial differential of interaction potential energy with respect to element entity density:

Finally, substituting formula (A-62) into formula (A-15) is the element sensitivity of the second element density:

The element sensitivity of the third element density is derived as follows:

Among them, the partial differential of the rigid matrix of the element to the solid density of the element can be obtained through formula (A-2) and formula (A-6):

Substituting Equation (A-65) into Equation (A-64) can obtain the partial differential of interaction potential energy to element entity density:

Finally, substituting formula (A-66) into formula (A-15) is the element sensitivity of the third element density:

The topological optimization method with multiple boundary conditions will be further described in detail below: the interaction potential energy mentioned earlier as the objective function is to describe the flexibility of the object when it is subjected to two forces, so the interaction potential energy can only be used for a single An input is computed with a corresponding single output. However, in the design of flexible mechanisms, sometimes a single flexible mechanism has multiple inputs and outputs. The design of multiple boundary conditions can be split into multiple boundary conditions with a single input and output for this setting. , respectively calculate their objective functions and then use linear superposition as the final objective function. Two situations will be described below. The first situation is that there is one input corresponding to two outputs in a single design interval, as shown in FIG. 13 . At this time, it can be divided into two boundary conditions, namely, the boundary condition that the input terminal corresponds to the output terminal #1, and the boundary condition that the input terminal corresponds to the output terminal #2, as shown in FIG. 14A and FIG. 14B . Calculate the objective functions of the two boundary conditions separately and then combine the weights to form the final topology optimization objective function as follows: f = f _{c 1} +ω ₁ f _{c 2} (A-68)

Among them, f _{c 1} and f _{c 2} are the objective functions calculated by boundary condition 1 and boundary condition 2 respectively, and ω ₁ is to adjust the weight between the two objective functions, because the factors that affect the final design result are f _{c 1} and The ratio between f _{and c 2} , so it is only necessary to add weights to the objective functions other than the first boundary condition for adjustment.

The second case is that there are two inputs and outputs in a single design interval, as shown in Figure 15. At this time, it can be split into two boundary conditions, namely input #1 corresponding to output #1 and output #2, and input #2 corresponding to the boundary conditions of output #1 and output #2, such as Figure 16A and Figure 16B. Similarly, the objective functions of the two boundary conditions are calculated separately and then combined with weights to form the final topology optimization objective function as shown in formula (A-68).

In addition to the two ways of splitting boundary conditions introduced above, the combination of the above two situations may also be encountered in the actual topology optimization of flexible mechanisms, that is, there are multiple inputs and multiple outputs, and there are A specific input corresponds to multiple outputs at the same time. In this case, the above two methods of splitting boundary conditions can also be combined, and weights can be added between multiple boundary conditions to manipulate the importance of different boundary conditions.

After adding the robust topology optimization, multi-material condition design and multi-boundary condition design into the SIMP method topology optimization, the original topology optimization process must be rewritten as shown in Figure 17. The detailed process is as follows:

Step 1: Define the design interval, boundary conditions and topology-related parameters.

Step 2: Use formula (A-13) and formula (A-14) to implement filtering algorithm and critical projection method for element density.

Step 3: Use the formula (A-14) to calculate the element entity density of the original design, the expanded design and the reduced design respectively (using different threshold values η).

Step 4: Carry out finite element analysis between different boundary conditions for the three designs respectively.

Step 5: Calculate the objective function of the problem equation for the three designs in a linear superposition manner.

Step 6: Use formula (A-18) to calculate the element sensitivities of the three designs respectively.

Step 7: Utilize the MMA method to update the element densities of the three materials in the three designs.

Step 8: Determine whether β and other related parameters need to be updated, and update if so.

Step 9: Determine whether it is converged, if yes, end; otherwise, skip to step 2 for the next iteration.

The flow of the topology optimization judgment condition in the program code is shown in FIG. 18 . Among them, loopbeta is the number of iterations of a single β value, loop is the number of iterations in the process of topology optimization, and iter is the number of iterations in the update of MMA variables. The detailed program structure flow is as follows:

Step 1: Define relevant parameters.

Step 2: Enter the while loop and start iterating until the convergence condition is met.

Step 3: Complete the filtering algorithm, critical projection method, finite element analysis, element sensitivity calculation and MMA variable update in sequence.

Step 4: Determine whether the parameters need to be updated. If the if judgment formula is met, β will be doubled, loopbeta and iter will be set back to 0, and change will be set back to 1.

Step 5: Judging whether it is converged, if β has increased to 512, element density change rate is less than 0.01 and the tolerance error of volume ratio is less than 0.001, jump out of the while loop and end the iteration; otherwise set loopbeta = loopbeta +1, loop=loop+ 1 and iter = iter +1 and go back to step three.

The following will further explain how the above design method is actually used to design an adaptive flexible gripper:

The section on multi-material conditions:

In this embodiment, additive manufacturing (AM) technology (that is, 3D printing) will be used for fabrication in the future. The material can be thermoplastic elastomer (thermoplastic elastomer, TPE), because TPE material has the elastic properties of vulcanized rubber and is Elastic deformation properties of plastics at high temperatures, but other flexible materials can also be used.

In order to facilitate 3D printing, the three materials used in this embodiment are formed by utilizing the mechanical properties (such as equivalent Young's modulus and density) of the same material at different filling densities. In detail, the shell layer is formed first, and then distributed in the shell layer with a predetermined pattern of texture (such as checkered pattern, honeycomb pattern, cross check pattern, etc.) and filling density. That is to say, the so-called multi-material in the present invention can cover the situation that the material is essentially the same kind but the mechanical properties are substantially different from each other in terms of interpretation, but the implementation is not limited to this, and materials of different nature can also be directly used (such as Silicone, TPU material, PP material, PE material, etc.).

In this example, TPE with a filling density of 40% and 100% is selected as the low-rigidity and high-rigidity materials, and TPE with a filling density of 80% is selected as a material with moderate rigidity to provide flexibility in the use of materials. Details See the material parameter table for three-material design:.

Three-material design material parameter table:

In addition, the volume constraint formula in the topology optimization process is formula (A-69):

為目標體積率，

、

、

則分別為材料一、材料二及材料三之體積率。材料一、材料二及材料三之目標體積率之總和需為原始設定之目標體積率。而在滿足式(A-69)之條件下，材料一、材料二及材料三之目標體積率則無額外之限制，意即最終之設計結果仍可全部使用三種材料之任一種。 in,

is the target volume ratio,

,

,

are the volume ratios of material 1, material 2 and material 3 respectively. The sum of the target volume ratios of Material 1, Material 2, and Material 3 must be the original target volume ratio. Under the condition of satisfying formula (A-69), there is no additional restriction on the target volume ratios of material 1, material 2, and material 3, which means that any of the three materials can still be used in the final design result.

The section on Multiple Boundary Design Conditions:

In the method of the self-adaptive flexible gripper of this embodiment, the design interval is further cut to form a plurality of design intervals that are related to each other, as shown in FIG. 19 . Among them, four rectangles (10)(20)(30)(40) respectively represent four design intervals, which are equivalent to four knuckles, but the implementation is not based on This is the limit, and it can also be three or more than four knuckles. The design goal is to apply the input displacement behind the first knuckle, and each knuckle can lean against each other after touching the grasped object. Pushing creates a bend that wraps around the object (A), as shown in Figure 20.

As shown in Figure 19, in detail, the self-adaptive flexible finger (1) of this embodiment includes a terminal phalanx (10), at least one middle phalanx (20) (30), a first phalanx (40) and a displacement transmission module (50), the displacement transmission module (50) includes a displacement transmission structure (51) and an actuator (52), the displacement transmission structure (51) is connected to the first phalanx ( 40), the two ends of the displacement transmission structure (51) (also called the moving end and the fixed end) are respectively fixed to a moving part (521) and a fixed part (522) of the actuator (52), the moving The member (521) moves towards the fixed member (522) through an actuator (523) (such as a screw, telescopic assembly, etc.), which can cause the displacement transmission structure (51) to generate extrusion and then transmit displacement and energy to achieve Gripping strategy shown.

In addition, a pair of hinge structures (60A) ( 60B) are connected. The first knuckle (40), the middle knuckle (20) (30) and the end knuckle (10) jointly define a clamping surface (100) on the same side, and one of the hinge structures (60A) is located adjacent At the clamping surface (100), another hinge structure (60B) is located relatively away from the clamping surface (100).

In order to achieve the gripping strategy shown in Figure 20 for the gripper jaws, the boundary conditions adopted in the design intervals of the knuckles (10)(20)(30)(40) can be referred to Figure 21A, Figure 21B, Figure 21C and Figure 21D respectively. The boundary conditions adopted in the design interval of the terminal phalanx (10) are shown in Figure 21D, where k _in and k _out are the spring constants of the input and output springs respectively, and f _in and f _out are the unit input and output power; the boundary conditions adopted in the design intervals of the first phalanx (40) and the middle phalanx (20) (30) are then shown in Figure 21A, Figure 21B, and Figure 21C respectively, wherein k _in , k _{out 1} and k _{out 2} are the spring constants of the springs at the input end, the first output end and the second output end respectively, k is the spring constant of the spring attached to the first output end in the x-axis direction, and f _in , f _{out 1} and f _{out 2} are respectively is the unit input and output force applied to the input terminal, the first output terminal and the second output terminal. Except for the terminal knuckles, the other knuckles have one input and correspond to two outputs at the same time. The goal is to maximize the displacement of the two output terminals at the same time under the given input displacement condition of the input terminal. The output direction of the first output end is downward, which can make the jaws approach the object to be gripped and then cover the object. The output direction of the second output end is leftward, and this output displacement can be used as the input displacement of the next knuckle To move the next knuckle, so the input displacement of the next knuckle is the displacement of the second output end of the previous knuckle. In addition, an additional spring can be provided in the horizontal direction of the first output end so as to be close to the setting of connecting with the next knuckle. In the boundary condition setting of the terminal phalanx, because the terminal phalanx does not need to push the next phalanx, there is only one input and one output, and the goal is to maximize the output under the given input displacement of the input terminal End displacement, so as to achieve the function of covering the object at the end of the jaw. The boundary conditions of the displacement transfer structure (51) are shown in Figure 22, which has an input port corresponding to an output port. The lower right part of the design interval is connected to the moving part of the jaw actuator, so the y direction of the node at this point The degrees of freedom are restricted, and the displacement output from the output terminal is used as the input displacement of the first knuckle. The node in the lower left corner is connected to the fixed part of the actuator, so the degrees of freedom of the node in the x and y directions are restricted.

After the above settings are completed, the topological optimization method will be used to design each knuckle and displacement transfer structure. Except for the terminal knuckle and the displacement transfer structure, the boundary conditions of the other knuckles will be further divided into two boundary conditions as shown in Figure 23A and Figure 23B. Respectively, the boundary condition corresponding to the first output terminal from the input terminal and the boundary condition corresponding to the second output terminal from the input terminal, respectively calculate their interaction potential energies and combine them in a linear superposition manner. Therefore, the objective functions of the middle and first knuckles are both shown in formula (A-70). Among them, MPE _{c 1} is the interaction potential energy obtained by taking Fig. 23A as the boundary condition, MPE _{c 2} is the interaction potential energy obtained by taking Fig. 23B as the boundary condition, ω ₁ is the adjusted ratio of MPE _{c 1} and MPE _{c 2} The weight value. The objective functions of the terminal phalanx and the displacement transfer structure are the interaction potential energy obtained by using Figure 21D and Figure 22 as the boundary conditions respectively, and the objective functions are shown in formula (A-71).

max: f = MPE (A-71)

Design interval and parameter setting: The design interval of each knuckle can be roughly rectangular, and the size can be set to have the same width, while the height decreases in sequence. Here, the height dimension can decrease sequentially from a maximum of 30mm and 40mm, and the width can be 20mm and 28mm, and the bottom of each knuckle can have a solid design area with a thickness of 3mm as a clamping surface.

The boundary condition settings of knuckles except the terminal knuckles all have one input corresponding to two outputs at the same time, so they are further divided into boundary conditions as shown in Figure 23A and Figure 23B, and the interaction potential energies are calculated respectively Then add them with the weight ω ₁ , and the topological optimization objective function is as follows:

The boundary conditions of the terminal knuckles are set as shown in Figure 21D, and the calculation of the topological optimization objective function is as follows:

The boundary conditions of the displacement transfer structure are shown in Figure 22. Since the displacement transfer structure has only one input corresponding to one output like the terminal knuckle, its topology optimization objective function is also shown in Equation (A-73).

In order to maintain the clarity of the final design results, each design interval is divided into about 10,000 to 20,000 elements, and the parameters related to topology optimization are shown in the "Topology Optimization Design Parameter Table" , the filtering radius can be set to be the side length of 5 unit elements, the Poisson's ratio can be set to 0.45, and in the step of implementing the critical projection method, the threshold values of the reduced, original and expanded designs can be set to 0.7, 0.5 and 0.3.

Topology optimization design parameter table:

The best design result obtained through the above-mentioned design conditions, the specific pattern of the self-adaptive flexible finger (1) of the present embodiment is as shown in Figure 24, each phalanx (10) (20) (30) (40) roughly It is a trapezoidal frame with multiple frames, and the displacement transmission structure (51) is roughly L-shaped. Wherein the displacement transmission mechanism (51), the first phalanx (40), the middle phalanx (20) (30) and the end phalanx (10) mainly adopt the first kind of flexible material (M01), the Both the first phalanx (40) and the middle phalanx (20) (30) adopt a second flexible material whose strength is relatively smaller than that of the first flexible material (M01) at the clamping surface (100). Material (M02). Specifically, the frame portion forming the clamping surface (100) includes the second flexible material (M02), and the first knuckle (40) is adjacent to the displacement transmission mechanism (51) The frame portion comprises the second flexible material (M02).

Preferably, between the first flexible material (M01) and the second flexible material (M02) can be filled with a third flexible material (not shown in the figure due to the small addition ratio) as a buffer, And the material strength of the third flexible material is between the first flexible material (M01) and the second flexible material (M02). In this embodiment, the first flexible material is the above material one (TPE filling rate 100%), the second flexible material is the above material three (TPE filling rate 40%), the third flexible material The material is the above material two (TPE filling rate 80%). In addition, the displacement transmission mechanism (51) can use a second flexible material at a part away from the fixed end, and the second flexible material can be the above-mentioned material three (TPE filling rate 40%), and the terminal knuckle (10) A second flexible material is used at a part of the inner frame edge relative to the clamping surface (100), and the second flexible material can be the above-mentioned material three (TPE filling rate 40%).

The self-adaptive flexible finger is tested for a number of related performances, and the A. input force, B. output displacement and C. output force of the self-adaptive flexible finger are tested under a given input displacement setting.

A. The experimental method of input force is to set the adaptive flexible finger on the linear moving platform, and push the adaptive flexible finger through the force sensor to give the input displacement, and during the movement of the adaptive flexible finger Record the input force displayed by the force sensor.

The experimental results of the input force are shown in the "Input Force Result Table".

Enter the force results table:

B. The experimental set-up method of the output displacement test is the same as that of the input force test, and an additional graph paper is set on the right side of the linear moving platform to record the position of the terminal knuckle output end of the adaptive flexible finger.

The experimental results of the output displacement are shown in the "Output Displacement Result Table".

Output displacement result table:

C. The experimental method of the output force test is to give the adaptive flexible finger an input displacement and record the output force under the condition that the output end is fixed (set a force sensor at the output end to record the output force of the adaptive flexible finger) . The test methods can be divided into experiment method 1 and experiment method 2. The difference between experiment method 1 and experiment method 2 is that in experiment method 1, the input displacement is given from the original state of the adaptive flexible finger, while in experiment method 2, the After giving the adaptive flexible finger an input displacement of 20mm in the opposite direction, take this state as the initial position, and then start to give the adaptive flexible finger an input displacement.

The experimental results of the output force are shown in the "Table of Output Force Result of Experimental Method 1" and "Table of Output Force Result of Experimental Method 2".

Experimental method 1 output force result table:

Output power result table of experimental method 2:

The self-adaptive flexible gripper (as shown in Figure 25 ) constituted by the self-adaptive flexible finger (1) of this embodiment is assembled on the actuator of horizontal linear movement, and the commercially available from Adaptive flexible grippers (model: Adaptive gripper finger DHAS) were tested and compared: the commercially available adaptive flexible grippers of FESTO Company, which can be used to grip objects with a gripping surface of about 95 mm and a thickness of 1 to 2 cm, while The clamping surface of the self-adaptive flexible finger (1) of the present embodiment is 83mm. Here, the grippers of FESTO Company are assembled on the horizontal linear movement actuator, and the actuator adopts the servo electric cylinder controller of TOYO Company, the model is TC-100. With the connecting jig, the distance between the ends of the jaws is 76mm when the actuator expands outward to the limit position, and the distance between the ends of the jaws is 18mm when it is compressed inward to the limit position.

In order to compare the adaptability of the grippers of this embodiment and the grippers developed by FESTO when gripping objects, four objects with different shapes were selected for gripping experiments. The four objects were circular workpieces, Square workpieces, trapezoidal objects and elastic bodies for engine feet. The test results are shown in the "Adaptiveness Performance Comparison Table". Here, the length of the gripper that is in contact with the surface of the object when gripping the object is used as the quantitative standard for self-adaptation. Based on this standard, it can be observed that the gripper of this embodiment is better at gripping The self-adaptability of the four kinds of objects is better than that of the grippers of FESTO company. When clamping a square workpiece, the clamping jaws of this embodiment can effectively cover it and successfully clamp it. The clamping jaw developed by FESTO cannot cover it or successfully clamp it. When clamping trapezoidal objects and engines When the foot is made of elastic body, the clamping jaw of this embodiment can make the clamping surface completely attached to the surface of the object to increase the stability during the clamping process, while the clamping jaw developed by FESTO can successfully clamp two objects but cannot clamp The clamping surface of the claw is attached to the surface of the object, and can only be gripped by contacting a specific position of the object.

Adaptive performance comparison table:

The grippers of this embodiment are better than the grippers developed by FESTO in adaptability for the following reasons. First, the displacement input method of the grippers is different. After touching the object, it can continue to transmit the displacement to each knuckle to fit the object Surface, secondly, the deformability of the structure of the gripper of this embodiment is better than that of the gripper developed by FESTO, so it is more able to adapt to the surface of objects with different shapes.

In terms of the load test of the grippers of FESTO, the test method is to gradually increase the weight of the lower tray after the grippers hold the object until the object slides down. The maximum weight that the jaws developed by FESTO can carry is 2.64 kilograms, and the maximum weight that the jaws of this embodiment can carry is about 6.71 kilograms. From the comprehensive experimental results, it can be seen that the adaptive flexible gripper of this embodiment is superior to the commercially available FESTO adaptive flexible gripper in terms of adaptive performance and maximum load.

It should be added that, as shown in Figure 25, although the above-mentioned self-adaptive flexible gripper is an example of a pair of self-adaptive flexible fingers (1), it is not limited to this, as shown in Figure 26, Figure As shown in 27, the self-adaptive flexible gripper can be composed of three, four or more self-adaptive flexible fingers (1) distributed in a ring. Or as shown in FIG. 28 , the adaptive flexible gripper can be composed of more than one pair of adaptive flexible fingers ( 1 ), the main purpose of which is to use a plurality of adaptive flexible fingers ( 1 ) to jointly define the clamping space.

It should be noted that the above content is only a preferred embodiment of the present invention, the purpose is to enable those skilled in the art to understand and implement the present invention, and is not used to limit the scope of the patent application of the present invention; therefore, it relates to the present invention All equivalent changes or modifications are covered by the scope of the patent application.

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Claims (10)

An adaptive flexible finger design method for multi-material condition design, to perform topology optimization through a computer implementation of a SIMP method (Solid isotropic material with penalization method), including the following steps: Step 1: Obtain the design interval, boundary conditions and Topological optimization of relevant parameter information, the design interval is at least a part of an adaptive flexible finger, and the initial value of the element density ρ _i of a plurality of materials is obtained, and the subscript i represents the i- th in the design interval Elements, wherein the plurality of materials are the same material but the mechanical properties are substantially different from each other; step 2: filter the element density ρ _i of various materials respectively, so as to convert the element density ρ _i of various materials into elemental entity density

;Step 3: Perform finite element analysis to obtain the node displacement, where the element solid density

The relationship with Young's coefficient E _i is:

in

,

in

Represents the physical density of the first element, the superscript p is the penalty coefficient, E _min is the minimum value close to zero, M is the amount of materials used plus 1, and Π is the multiplication operator; step 4: use the node Displacement calculation objective function and element sensitivity; step 5: update the element density of various materials; step 6: judge whether it is converged, if it converges, it will end; otherwise, go to step 2 for the next iteration. The adaptive flexible finger design method for multi-material condition design as described in Claim 1, wherein the volume constraint formula in the SIMP method is:

; where v _i is the volume of the element, V is the volume of the design interval,

is the target volume ratio. The adaptive flexible finger design method for multi-material conditional design as described in claim 1, wherein the element density ρ _i of various materials is first converted into an intermediate element density by an element density filtering algorithm

, and then converted to element entity density by a critical projection method

; The calculation formula of the element density filtering algorithm is as follows:

,M =1 , ... n, n >1; where,

is the density of the Mth intermediate element, N _e,i is the set of elements within the filtering radius around the i-th element; the subscript j means that the element is the jth element in N _e,i ; w _ij is the filtering weight function; v _j is the volume of the element; the calculation formula of the critical projection method is as follows:

,M =1,... n, n >1; where,

is the entity density of the Mth element, β is a projection parameter selected from 1 to 512, and η is a threshold value between 0 and 1. The self-adaptive flexible finger design method of multi-material conditional design as described in claim 3, wherein, the SIMP method is to add a robust topology optimization as follows, the original design, expansion design and reduction design of the objective function Also minimize:

Among them, the superscript and subscript d , o and e represent the expanded design, the original design and the reduced design respectively, K is the global rigidity matrix, U is the global displacement vector, and F is the global force vector. The adaptive flexible finger design method of multi-material condition design as described in claim 4, wherein, the SIMP method uses mutual potential energy (Mutual potential energy, MPE ) as the objective function, and the adaptive flexible finger includes a terminal phalanx, a first phalanx, at least one middle phalanx and a displacement transmission structure, the middle phalanx is located between the terminal phalanx and one end of the first phalanx, the displacement transmission structure connects the first The other end of the phalanx, wherein: the topology optimization objective function f of the first phalanx and the middle phalanx is:

The topological optimization objective function f of the terminal phalanx and the displacement transfer structure is: max: f = MPE , min: f =- MPE , where:

; MPE _{c 1} is the interaction potential energy obtained from the boundary conditions corresponding to a first output end from an input end;

is the interaction potential energy obtained from the boundary condition corresponding to the input terminal to a second output terminal; ω ₁ is the weight value; K is the global rigid matrix including the spring constants connected to the input terminal and the output terminal; U ₁ is the input load only The global displacement vector under; U ₂ is the global displacement vector under only the output load,

The superscript T in is the transposition matrix, and the fixed end displacement in the global displacement vector is set to 0. An adaptive flexible finger designed with multi-material conditions is produced by using the adaptive flexible finger design method for multi-material conditional design as described in claim 5, using flexible materials, including the displacements connected in sequence transfer mechanism, the first phalanx, the middle phalanx and the terminal phalanx. An adaptive flexible finger designed with multi-material conditions, comprising a displacement transmission mechanism, a first phalanx, at least one middle phalanx and a terminal phalanx connected in sequence, the first phalanx, the middle phalanx and the terminal knuckles are generally trapezoidal frame with multiple frame parts, the displacement transmission mechanism, the first knuckle, the middle knuckle and the terminal knuckle mainly adopt the first kind of flexible material, the first knuckle The phalanx, the middle phalanx and the terminal phalanx all form a clamping surface together on the same frame portion, forming The frame portion of the clamping surface includes a second flexible material whose material strength is relatively smaller than that of the first flexible material, and the frame portion adjacent to the displacement transmission mechanism of the first knuckle includes the A second flexible material, wherein the first flexible material and the second flexible material are the same material but have substantially different mechanical properties from each other. An adaptive flexible gripper, comprising a plurality of adaptive flexible fingers designed with multi-material conditions as described in any one of claim 6 to claim 7, and the plurality of adaptive flexible fingers jointly define a grip holding space. A computer program product, which stores a program. When the program is loaded into the computer and executed, it can complete the self-adaptive flexible finger design of the multi-material condition design as described in any one of item 1 to item 5 of the request method. A computer-readable recording medium, which stores a program. After the computer loads and executes the program, it can complete the self-adaptive flexible design of multi-material conditions as described in any one of items 1 to 5 of the request. Sex finger design method.

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