Modeling method for stiffness of variable cross-section robotic arm links in relation to structural dimensions

Abstract

This invention relates to a method for modeling the stiffness of a variable cross-section robotic arm link in relation to structural dimensions. The method includes the following steps: Step 1: Dividing the robotic arm link into a connecting rod section and front and rear connecting sections; Step 2: Equipping the connecting rod section as an Euler-Bernoulli beam and deriving its stiffness matrix using the principle of virtual work. The connecting rod section model is divided into tension / compression, bending, and torsional stress modes. The stiffness matrix for each stress mode is calculated, and the bending stress mode matrix is ​​corrected before being integrated with other stress mode matrices to obtain the connecting rod section stiffness matrix K2. The parameters in K2 are then correlated with the actual cross-sectional dimensions of the connecting rod section; Step 3: Extracting the stiffness matrices K1 and K3 of the front and rear connecting sections using finite element analysis software; Step 4: Converting and superimposing K1, K2, and K3 to obtain the robotic arm link stiffness matrix; Step 5: Using this method to guide the design of the robotic arm link. The stiffness model established by this invention is accurate and fast in calculation, facilitating dimensional optimization and quickly obtaining design dimensions that meet stiffness requirements.

Description

Technical Field

[0001] This invention relates to the field of industrial robot design, specifically a method for modeling the stiffness of a variable cross-section robotic arm link in relation to structural dimensions. Background Technology

[0002] The stiffness of a robotic arm represents its ability to resist deformation under external loads, and it has a significant impact on the working accuracy of the robotic arm. Therefore, understanding the stiffness of a robotic arm and increasing its stiffness while meeting requirements can improve its ability to resist external loads, thereby effectively improving its accuracy. This is of great significance in the design phase of a robotic arm. Generally, the study of robotic arm stiffness can be divided into two parts: joint modules and link modules. Among them, the link modules only contain structural components, making it easier to establish a stiffness model, and they also form the basis for establishing the overall stiffness model of the robotic arm.

[0003] A common method for stiffness modeling of robotic arm links is to use commercial finite element software. This method produces a very accurate stiffness model, but it is very time-consuming, especially in the design phase. The link dimensions need to be changed frequently, and each time the parameters are changed, the mesh needs to be re-generated and analyzed. Overall, the cost is high and the computational efficiency is low.

[0004] Compared to the previous method, using structural mechanics to establish the stiffness matrix of the link provides a comprehensive and thorough understanding of the influence of each design dimension of the link on its stiffness. Compared to the finite element method, it has a shorter computation time, and the fully parameterized model is easier to modify. However, the robotic arm link is a variable cross-section link, unlike the regularly shaped cross-section links in general methods. Therefore, general structural mechanics methods are not applicable. When deriving the stiffness matrix of the link using structural mechanics methods, the stress situation of the link is considered, and it can be regarded as a spatial beam model before calculating its stiffness. However, for the robotic arm link, due to the complex structure at both ends of the link, it is impossible to establish an analytical model. Summary of the Invention

[0005] The purpose of this invention is to provide a method for modeling the stiffness of a variable cross-section robotic arm link that is related to structural dimensions. The stiffness model established by this method is accurate and fast, which not only facilitates the subsequent dimensional optimization of the robotic arm link, but also allows for the rapid acquisition of robotic arm link design dimensions that meet performance requirements.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] A method for modeling the stiffness of a variable cross-section robotic arm link in relation to structural dimensions includes the following steps:

[0008] Step 1: Construct an analysis model and divide the entire robotic arm linkage into a front connection part, a link part, and a rear connection part, where the link part is a thin-walled box-shaped structure;

[0009] Step 2: Model the connecting rod section and solve for its stiffness matrix K2. Specifically:

[0010] Step 2.1: Equivalently represent the connecting rod section as an Euler-Bernoulli beam;

[0011] Step 2.2: Determine the stiffness matrix of the connecting rod using the principle of virtual work. The derivation formula is as follows:

[0012] (1);

[0013] In equation (1) above, D is the elastic matrix and B is the strain transformation vector;

[0014] Step 2.3: Divide the model of the connecting rod (2) into three stress modes: tension and compression stress mode, bending stress mode and torsion stress mode. First, obtain the elastic matrix D and strain conversion vector B under each stress mode, and then substitute them into equation (1) to obtain the stiffness matrix under each stress mode.

[0015] Step 2.4 Since the connecting rod (2) is a thin-walled box structure, the connecting rod (2) is modeled separately under shear deformation to obtain the pure shear stress mode. The matrix under the pure shear stress mode is linearly superimposed with the matrix under the bending stress mode to achieve correction. Then, the stiffness matrix under the corrected bending stress mode is integrated with the stiffness matrix under other stress modes to finally obtain the stiffness matrix K2 of the connecting rod (2).

[0016] Step 2.5: Associate each parameter in the stiffness matrix K2 with the actual cross-sectional dimensions of the connecting rod (2);

[0017] Step 3: Perform stiffness modeling on the front connection part (1) and the rear connection part (3) respectively, and use finite element software to extract the stiffness matrix K1 of the front connection part (1) and the stiffness matrix K3 of the rear connection part (3);

[0018] Step 4: Transform K1, K2, and K3 to the same coordinate system and superimpose them to obtain the stiffness matrix of the entire robotic arm linkage. ;

[0019] Step 5: Using the stiffness matrix Provide guidance on the design of robotic arm linkages.

[0020] In step 2.3, the strain transformation vector B for the tension / compression mode and the torsional mode is:

[0021] (2);

[0022] The strain transformation vector B of the bending stress mode is:

[0023] (3);

[0024] In equation (3) above, x and y are the x-axis and y-axis coordinates along the connecting rod (2), respectively. l Let N be the length of the connecting rod (2), and N be a shape function, where N is the force for tension / compression mode and torsion mode:

[0025] (4);

[0026] The N value for the bending stress mode is:

[0027] (5);

[0028] In equations (4) and (5) above:

[0029] ;

[0030] Where x is the absolute coordinate of the displacement field of the element along the x-axis of the connecting rod; s is the relative coordinate of the displacement field of the element along the x-axis of the connecting rod.

[0031] The elastic matrix D under tension / compression and bending stress modes is as follows:

[0032] (6);

[0033] In equation (6) above, E is Young's modulus;

[0034] The elastic matrix D under torsional stress mode is:

[0035] (7);

[0036] In the above equation (7), G is the shear modulus and μ is Poisson's ratio;

[0037] Substituting equations (2)-(7) into equation (1) yields the stiffness matrix under each stress mode, where:

[0038] The stiffness matrix under tension and compression modes is:

[0039] (8);

[0040] In the above formula (8), A is the cross-sectional area of ​​the connecting rod portion; l The length of the connecting rod portion;

[0041] The stiffness matrix under torsional stress mode is:

[0042] (9);

[0043] In the above formula (9), J is the torsional moment of inertia of the connecting rod.

[0044] The stiffness matrix under bending stress mode is:

[0045] (10);

[0046] In the above formula (10), It represents the bending moment of inertia of the connecting rod about the y-axis or about the z-axis.

[0047] In step 2.4, the parameters under the pure shear stress mode are as follows:

[0048] Shape function N:

[0049] (12);

[0050] Strain transformation vector B:

[0051] (13);

[0052] Elasticity matrix D:

[0053] (14);

[0054] In the above formula (14), For cross-sectional area, The shear coefficient of the cross section;

[0055] Substituting equations (13) and (14) into equation (1), we obtain the stiffness matrix for the pure shear mode:

[0056] (15);

[0057] The stiffness matrix under the modified bending stress mode is obtained by superimposing the stiffness matrix (15) of the pure shear mode and the stiffness matrix (10) of the bending stress mode:

[0058] (16);

[0059] The elements in equation (16) above are as follows:

[0060]

[0061] ;

[0062] Integrating equations (16), (8), and (9) yields the final stiffness matrix of the connecting rod:

[0063] (17);

[0064] The expressions for each element in equation (17) above are:

[0065] ,

[0066]

[0067] .

[0068] The parameters in the stiffness matrix K2 of the connecting rod include the cross-sectional area of ​​the two end faces of the connecting rod, the moment of inertia of the cross-section of the two end faces of the connecting rod, the torsional moment of inertia of the two end faces of the connecting rod, and the shear coefficient of the cross-section of the two end faces of the connecting rod. In step 2.5, the parameters in K2 are related to the actual cross-sectional dimensions of the two end faces of the connecting rod as follows:

[0069] The cross-sectional areas of the two ends of the connecting rod are:

[0070] (18);

[0071] In the above formula (18), The width of the outer wall of the cross section. t is the length of the outer wall of the cross section, and t is the thickness of the four walls of the cross section.

[0072] The moments of inertia of the cross sections at both ends of the connecting rod are:

[0073] (19);

[0074] (20);

[0075] The torsional moments of inertia of the two end faces of the connecting rod are:

[0076] (twenty one);

[0077] The coefficient β in equation (21) above is selected from a table;

[0078] Shear coefficients of the two end faces of the connecting rod:

[0079] (25);

[0080] In the above formula (25):

[0081] The parameters are ; The parameters are ;

[0082] and .

[0083] In step 2.5, the cross-sectional area of ​​equations (18)-(21) is... A、 Moment of inertia of cross section I、 Torsional moment of inertia of cross section J Perform the following equivalent processing:

[0084] Cross-sectional area A :

[0085] ;

[0086] Moment of inertia of cross section I :

[0087] ;

[0088] Torsional moment of inertia of cross section J :

[0089] .

[0090] In step three, the front and rear connecting parts are both regarded as super elements with two nodes using finite element software. One end node is fixed and the stiffness matrix of the other end node is extracted to obtain the stiffness matrix K1 of the front connecting part and the stiffness matrix K3 of the rear connecting part.

[0091] In step four, we first derive the coordinate transformation formula that transforms K1, K2, and K3 to the same coordinate system. T :

[0092] (26);

[0093] In the above formula (26):

[0094] ;

[0095] in Local coordinate systems representing the front connecting part, connecting rod part, and rear connecting part For the global coordinate system The direction cosine;

[0096] Then, transform each stiffness matrix to the same coordinate system:

[0097] (27);

[0098] The stiffness matrix of the front connection in the global coordinate system is obtained by using the above formula (27). Stiffness matrix of the connecting rod Stiffness matrix of the rear connection ;

[0099] Will , , Linear superposition yields the stiffness matrix of the entire robotic arm linkage. :

[0100] (28).

[0101] The advantages and positive effects of this invention are as follows:

[0102] 1. This invention targets the linkage of a variable cross-section robotic arm, dividing it into a front connection part, a link part, and a rear connection part for separate analysis. The link part is equivalent to a spatial beam model, and its analytical stiffness matrix is ​​established using the principle of virtual work. The front and rear connection parts are considered as superelements, and their stiffness matrices are obtained using finite element software. Then, the three stiffness matrices are transformed into linear superposition in the global coordinate system to obtain the stiffness matrix of the entire link. Since the link stiffness matrix is ​​related to the structural dimensions of the link part, it facilitates subsequent dimensional optimization. At the same time, design dimensions that meet performance requirements can be quickly obtained based on stiffness requirements. Furthermore, the stiffness of the connection parts at both ends is also taken into account, making the model more accurate.

[0103] 2. When performing analytical stiffness modeling on the connecting rod, this invention equates it to an Euler-Bernoulli beam and uses the principle of virtual work to reduce the amount of calculation. Furthermore, a new pure shear stress mode is added to correct the matrix and improve its accuracy. When solving for the parameters in the stiffness matrix, the equivalent method is used to further reduce the amount of calculation. The final overall stiffness matrix of the connecting rod can be calculated accurately, conveniently, and quickly. Attached Figure Description

[0104] Figure 1 This is a schematic diagram of the process of the method of the present invention.

[0105] Figure 2 This is a schematic diagram of the variable cross-section robotic arm linkage structure addressed in this invention.

[0106] Figure 3 This is a schematic diagram of the coordinate system for the Euler-Bernoulli beam model.

[0107] Figure 4 for Figure 2 Schematic diagram of the middle connecting rod section.

[0108] Figure 5 for Figure 2 Schematic diagram of the front and rear connecting parts.

[0109] Figure 6 This is a schematic diagram showing the relationship between the coordinate systems of the front connecting part, the rear connecting part, and the connecting rod part and the world coordinate system.

[0110] Among them, 1 is the front connecting part, 2 is the connecting rod part, and 3 is the rear connecting part. Detailed Implementation

[0111] The invention will now be described in further detail with reference to the accompanying drawings.

[0112] like Figures 1-6 As shown, the present invention includes the following steps:

[0113] Step 1: Construct an analysis model and divide the entire robotic arm linkage into a front connecting part 1, a link part 2, and a rear connecting part 3, where the link part 2 is as follows: Figure 4 The diagram shows a thin-walled box-type structure.

[0114] like Figure 2 As shown, this invention targets the linkage of a variable cross-section robotic arm, dividing it into a front connecting part 1, a connecting part 2, and a rear connecting part 3, which will be analyzed separately later.

[0115] Step 2: Model the connecting rod 2 and solve for the stiffness matrix K2 of the connecting rod 2.

[0116] The specific process for this step is as follows:

[0117] Step 2.1, first as follows Figure 3 As shown, considering all the force conditions of link 2, link 2 is equivalent to an Euler-Bernoulli beam. The force conditions of this model are consistent with the actual force conditions of link 2, both of which bear six-dimensional spatial forces. The coordinate system of this model is defined as follows: Figure 3 As shown in the figure, the coordinate system is positive. The model of the connecting rod part 2 has a node one and a node two at both ends, where node one corresponds to the front connecting part 1 and node two corresponds to the rear connecting part 3. Figure 3 In R and R represent the translational and rotational deformations along each axis, respectively.

[0118] Step 2.2: Derive the stiffness matrix of the connecting rod 2. There are two methods in structural mechanics: the principle of minimum potential energy and the principle of virtual work. The principle of minimum potential energy requires understanding variational principles, which is somewhat complex. This invention uses the principle of virtual work to derive the stiffness matrix of the connecting rod. The derivation formula is as follows:

[0119] (1);

[0120] In equation (1) above, k is the stiffness matrix we need to find; D is the elasticity matrix, which is related to the material properties; and B is the strain conversion vector.

[0121] The source of the above formula (1) can be found in "[1] Zhang Junfeng, Yin Huina, Li Jie, et al. Derivation of stiffness matrix of Euler-Bernoulli beam element [J]. Journal of Hydraulic and Architectural Engineering, 2019, 17(3):6".

[0122] Step 2.3: Divide the model of the connecting rod part 2 into three stress modes: tension-compression stress mode (tension-compression mode), bending stress mode, and torsion stress mode. First, obtain the elastic matrix D and strain conversion vector B under different stress modes, and then substitute them into equation (1) to obtain the stiffness matrix under each stress mode.

[0123] For different stress modes, the strain transformation vector B in equation (1) above is different, as follows:

[0124] The strain transformation vector B between the tension / compression mode and the torsional mode is:

[0125] (2);

[0126] The strain transformation vector B of the bending stress mode is:

[0127] (3);

[0128] In equation (3) above, N is the shape function. The shape function N varies for different modes, where:

[0129] The values ​​of N for tension / compression and torsion stress modes are:

[0130] (4);

[0131] The N value for the bending stress mode is:

[0132] (5);

[0133] Substituting the shape function N under different modes in equations (4) and (5) into equations (2) and (3) above, we can obtain the strain transformation vector B under different stress modes.

[0134] In the above formulas (2)-(5), x and y are the coordinates along the x-axis and y-axis, respectively. l The main body length of the beam ，s The following description is provided: Before proceeding to the next calculation, a coordinate transformation is required, using relative positions. Do not use absolute position The relationship between the two is as follows:

[0135] ;

[0136] In the above formula, x is the absolute coordinate of the element displacement field along the x-axis of the connecting rod; s is the relative coordinate of the element displacement field along the x-axis of the connecting rod. This transformation facilitates subsequent solutions.

[0137] Furthermore, in the above equation (1), the elastic matrix D is different under different force modes, as follows:

[0138] The elastic matrix D under tension / compression and bending stress modes is as follows:

[0139] (6);

[0140] In equation (6) above, E is Young's modulus and matrix D is a square matrix whose size is related to the strain transformation vector.

[0141] The elastic matrix D under torsional stress mode is:

[0142] (7);

[0143] In the above formula (7), G is the shear modulus and μ is Poisson's ratio, which is a fixed value related to the material. Different materials have different Poisson's ratios.

[0144] For equations (2)-(7) above, please refer to the following references: "[1] Dong Jun, Liu Xuhong, Yao Shunzhong, et al. Analysis of stiffness matrix of spatial beam element based on virtual work principle [J]. Journal of Southwest Forestry University, 2002, 022(004):59-62.", "Zeng Pan. Finite element analysis and application [M]. Tsinghua University Press, 2004.", "Zhang Junfeng, Yin Huina, Sun Dayong, et al. Derivation of stiffness matrix of Euler beam element considering shear deformation based on shape function [J]. Journal of Chongqing Jiaotong University: Natural Science Edition, 2020, 39(9):8."

[0145] Substituting the elastic matrix D and strain transformation vector B from equations (2)-(7) for different stress modes into the stiffness matrix calculation formula (1), we can obtain the stiffness matrix for each stress mode, specifically:

[0146] The stiffness matrix under tension and compression modes is:

[0147] (8);

[0148] In the above formula (8), A is the cross-sectional area of ​​the connecting rod part 2; l is the length of the connecting rod part 2.

[0149] The stiffness matrix under torsional stress mode is:

[0150] (9);

[0151] In the above formula (9), J is the torsional moment of inertia of the connecting rod 2.

[0152] The stiffness matrix under bending stress mode is:

[0153] (10);

[0154] In the above formula (10), It is the bending moment of inertia of the main body of the connecting rod about the y-axis or about the z-axis.

[0155] Without considering shear deformation, the overall stiffness matrix of the connecting rod 2 can be obtained by integrating the stiffness matrices of equations (8)-(10) above:

[0156] (11);

[0157] like Figure 3 As shown, the connecting rod 2 has two nodes, node one and node two. The stiffness matrix should be 12×12. However, the matrix is ​​symmetric, and we usually fix one end node when applying it. Therefore, we only need one 6×6 submatrix, as shown in equation (11) above.

[0158] The matrix integration process described above is as follows:

[0159] For each force pattern, the following formula applies:

[0160] ;

[0161] in, It is the external force applied to both ends of the connecting rod under this force mode; It is the stiffness matrix of the link under each stress mode; It refers to the deformation at both ends of the connecting rod caused by external forces under this force mode.

[0162] Under tension and compression stress modes:

[0163] ;

[0164] ;

[0165] in, and The force applied along the x-axis to the connecting rod 2; and This represents the translational deformation along the x-axis generated at both ends of the connecting rod.

[0166] Torsional stress mode:

[0167] ;

[0168] ;

[0169] in, and The torque applied around the x-axis to both ends of the connecting rod; and This refers to the rotational deformation about the x-axis generated at both ends of the connecting rod.

[0170] Bending force mode around the y-axis:

[0171] ;

[0172] ;

[0173] in, and The force along the y-axis and the torque about the z-axis applied to both ends of the connecting rod; and The translational deformation along the y-axis and the rotational deformation about the z-axis generated at both ends of the connecting rod.

[0174] Bending force mode around the z-axis:

[0175] ;

[0176] ;

[0177] in, and The force along the z-axis and the torque about the y-axis applied to both ends of the connecting rod; and The translational deformation along the z-axis and the rotational deformation about the y-axis generated at both ends of the connecting rod.

[0178] By integrating the stiffness matrices from various modes, the resulting link stiffness matrix under any mode satisfies the following formula:

[0179] ;

[0180] Where P is the external force applied to both ends of the link under any force mode; K is the stiffness matrix of the link under each mode under any force mode. It refers to the deformation at both ends of the connecting rod caused by external forces under any stress mode.

[0181] The relationship is as follows:

[0182] ;

[0183] ;

[0184] Because the stiffness matrix is ​​highly symmetric, and because we typically fix one end of the link when solving for the stiffness matrix, i.e., restricting six degrees of freedom at one end of the link, the simplified link P and... for:

[0185] ;

[0186] ;

[0187] The stiffness matrix K can be determined based on the corresponding P and The integration process involves placing the necessary elements from the stiffness matrices of each stress mode in their appropriate positions. This integration process is a well-known technique in the field.

[0188] Step 2.4: Introduce the pure shear stress mode and linearly superimpose the matrices of the pure shear stress mode and the bending stress mode to achieve correction. Then, integrate the stiffness matrix of the corrected bending stress mode with the stiffness matrices of other stress modes to obtain the analytical stiffness matrix K2 of the connecting rod 2.

[0189] In the above calculation of the stiffness matrix, it is assumed that the shear deformation caused by bending has an extremely small effect on the beam deflection and can be ignored. This is consistent with general structures and simplifies the calculation process. However, if... Figure 4 As shown, the connecting rod part 2 analyzed in this invention is a thin-walled box beam structure, and the shape of the beam is similar to a short and thick beam. The shear deformation caused by bending cannot be ignored, and this part needs to be solved to correct the overall stiffness matrix.

[0190] The solution method in this step is to model the effect of shear deformation separately, called the pure shear stress mode. After modeling, it is linearly superimposed with the bending stress mode to obtain the stiffness matrix under the modified bending stress mode, specifically:

[0191] A new pure shear stress mode is introduced, and the parameters of this mode are shown below:

[0192] Shape function N:

[0193] (12);

[0194] Strain transformation vector B:

[0195] (13);

[0196] Elasticity matrix D:

[0197] (14);

[0198] In the above formula (14), is the junction shear coefficient, the value of which is given in step 2.5.

[0199] Substituting equations (13) and (14) into equation (1), we obtain the stiffness matrix for the pure shear mode:

[0200] (15);

[0201] In the above formula (15), Let be the cross-sectional area, the value of which is given in step 2.5.

[0202] Finally, by superimposing the stiffness matrix (15) of the pure shear mode and the stiffness matrix (10) of the bending mode, we can obtain the stiffness matrix of the modified bending mode:

[0203] (16);

[0204] The elements in equation (16) above are as follows:

[0205]

[0206] ;

[0207] After integrating the above equation (16) with the stiffness matrix equations (8) and (9) under other stress modes, the final stiffness matrix of the connecting rod part 2 is obtained after modification (see step 2.3 for the integration process):

[0208] (17);

[0209] The expressions for each element in equation (17) above are:

[0210] ,

[0211]

[0212]

[0213] .

[0214] Step 2.5: Associate each parameter in the stiffness matrix K2 with the actual cross-sectional size of the connecting rod 2. In this way, the parameters of the overall stiffness matrix K2 of the connecting rod 2 are solved by the length of the connecting rod 2 and the actual cross-sectional size. Then, the parameters are substituted into the overall stiffness matrix equation (17) of the connecting rod 2 to obtain the actual stiffness matrix K2 of the connecting rod 2 under this size.

[0215] The parameters in equation (17) are related to the actual cross-sectional dimensions of the two end faces of the connecting rod 2 as follows:

[0216] The structure of the connecting rod part 2 is as follows Figure 4 As shown, the cross-sectional areas at both ends are:

[0217] (18);

[0218] In the above formula (18), The width of the outer wall of the cross section. t is the length of the outer wall of the cross section, t is the thickness of the four walls of the cross section, and i=1,2 means that the parameters to be calculated are the parameters at both ends of the cross section. The thicker end of the connecting rod part 2 is regarded as the first end face, and the thinner part is regarded as the second end face.

[0219] The moments of inertia of the cross sections of the two end faces of the connecting rod 2 are:

[0220] (19);

[0221] (20);

[0222] In equations (19) and (20) above and These represent the moments of inertia of the cross section when bending about different axes.

[0223] The torsional moments of inertia of the two end faces of the connecting rod 2 are:

[0224] (twenty one);

[0225] The coefficients in equation (21) above The value can be selected by looking up a table.

[0226] After obtaining the geometric parameters of the two ends of the connecting rod 2, these parameters need to be equivalently processed to simplify subsequent calculations. The simplified parameters can be directly substituted into the stiffness matrix for solving. The simplified expression is as follows:

[0227] Cross-sectional area A :

[0228] (twenty two);

[0229] Moment of inertia of cross section I :

[0230] (twenty three);

[0231] Torsional moment of inertia of cross section J :

[0232] (twenty four);

[0233] The shear coefficients of the two end faces of the connecting rod 2 are:

[0234] (25);

[0235] In the above formula (25):

[0236] The parameters are ; The parameters are ;

[0237] and ;

[0238] After substituting the actual dimensions of the connecting rod 2, the above parameters can be obtained. Then, the obtained parameters are substituted into the overall stiffness matrix (17) of the connecting rod 2 to obtain the actual stiffness matrix K2 of the connecting rod 2.

[0239] In this step, since the stiffness matrix K2 is related to the actual size of the connecting rod 2, on the one hand, the size of the connecting rod 2 can be adjusted in real time to obtain the stiffness matrix K2 under different sizes for optimization and comparison; on the other hand, when the design stiffness is determined, the structural size of the connecting rod 2 that meets the stiffness requirements can also be obtained quickly.

[0240] For equations (18)-(25), please refer to the following documents: "[1] Zhang Junfeng, Li Jie, Yin Huina, et al. Stiffness matrix of variable cross section Euler beam element considering shear deformation [J]. Structural Engineer, 2020, 36(2):8." and "[1] Shi Binghua. Calculation formula of the non-uniformity coefficient of shear stress distribution of common cross sections [J]. Journal of Building Structures, 1984(02):66-70."

[0241] Step 3: Perform stiffness modeling on the front connecting part 1 and the rear connecting part 3 respectively, such as... Figure 5 As shown, by using finite element software, the front connecting part 1 and the rear connecting part 3 can be regarded as super elements with two nodes, and their stiffness matrix can be directly extracted.

[0242] Specifically: such as Figure 5 As shown, when processing with finite element software, the front connecting part 1 and the rear connecting part 3 are each regarded as a super element with two nodes (Node1 and Node2). One end node is fixed, and the stiffness matrix of the other end node is extracted. Finally, the stiffness matrix K1 of the front connecting part 1 and the stiffness matrix K3 of the rear connecting part 3 can be obtained. The front connecting part 1 corresponds to the thicker end of the connecting rod 2, and the rear connecting part 3 corresponds to the thinner end of the connecting rod 2. The finite element software is a well-known technology in this field.

[0243] Step 4: Perform coordinate transformation on the stiffness matrix K1 of the front connecting part 1, the stiffness matrix K2 of the connecting part 2, and the stiffness matrix K3 of the rear connecting part 3. Transform K1, K2, and K3 into the same coordinate system and superimpose them to obtain the stiffness matrix of the entire robotic arm link.

[0244] Specifically, the process involves first obtaining the coordinate transformation formula for converting K1, K2, and K3 to the same coordinate system. T :

[0245] (26);

[0246] In the above formula (26):

[0247] ;

[0248] in Local coordinate system representing front connecting part 1, connecting rod part 2, and rear connecting part 3 For the world coordinate system The direction cosine, and the relationship between the local coordinate system and the global coordinate system (world coordinate system) of each part are as follows: Figure 6 As shown.

[0249] Then, transform each stiffness matrix to the same coordinate system:

[0250] (27);

[0251] According to the above equation (27), the stiffness matrix of the front connection part 1 in the global coordinate system can be obtained. Stiffness matrix of link 2 Stiffness matrix of the rear connection part 3 ,Will , , Linear superposition yields the stiffness matrix of the entire robotic arm linkage. :

[0252] (28).

[0253] Step 5: Using the stiffness matrix Provide guidance on the design of robotic arm linkages.

[0254] Specifically, due to the stiffness matrix of the entire robotic arm linkage in this invention... Since the dimensions of the connecting rod 2 are related to its structure, the parameters of K2 in the overall stiffness matrix of the connecting rod 2 can be solved using the length and actual cross-sectional dimensions of the connecting rod 2, and then these parameters can be substituted into the stiffness matrix of the robotic arm link. Obtain the actual stiffness matrix of the robotic arm link at this dimension. By analyzing different sizes The comparison enables the optimization of the dimensions of the robotic arm links. Furthermore, after determining the stiffness requirements of the robotic arm, design dimensions that meet the performance requirements can be quickly obtained based on the stiffness. This invention also considers the stiffness of the connecting parts at both ends, making the overall stiffness model more accurate.

[0255] The stiffness of the links in a robotic arm is crucial to the overall stiffness of the robotic arm. Stiffness models can be used for structural optimization and vibration suppression, making the establishment of stiffness models for robotic arm links extremely important.

[0256] The mechanical arm linkage stiffness model established in this invention Unlike existing technologies:

[0257] First, the objects are different. The object of analysis in this invention is a variable cross-section connecting rod, and the connecting rod part 2 is a thin-walled box structure. The linear change in the cross-sectional dimensions of the object makes the analysis difficult.

[0258] Secondly, the types of models are different. The linkage stiffness model established in this invention is a combination of the finite element model of the two-end connection part and the structural mechanics analytical model of the main linkage part 2.

[0259] Furthermore, the model of the connecting rod 2 in this invention is related to the main structural dimensions of the connecting rod 2. When performing analytical stiffness modeling on the connecting rod 2, it is equivalent to an Euler-Bernoulli beam and the principle of virtual work is used to reduce the amount of calculation. In addition, a new pure shear stress mode is added to correct the matrix and improve its accuracy. When solving the parameters in the stiffness matrix, the equivalent method is used to further reduce the amount of calculation. Finally, the overall stiffness matrix of the connecting rod 2 is both accurate and easy to calculate.

Claims

1. A method for modeling the stiffness of a variable cross-section robotic arm link in relation to structural dimensions, characterized in that: Includes the following steps: Step 1: Construct an analysis model and divide the entire robotic arm linkage into a front connecting part (1), a connecting part (2) and a rear connecting part (3), wherein the connecting part (2) is a thin-walled box-shaped structure; Step 2: Model the connecting rod (2) and solve for the stiffness matrix K2 of the connecting rod (2), specifically: Step 2.1: Equivalently represent the connecting rod (2) as an Euler-Bernoulli beam; Step 2.2: Determine the stiffness matrix of the connecting rod (2) using the principle of virtual work. The derivation formula is as follows: (1)； In equation (1) above, D is the elastic matrix and B is the strain transformation vector; Step 2.3: Divide the model of the connecting rod (2) into three stress modes: tension and compression stress mode, bending stress mode and torsion stress mode. First, obtain the elastic matrix D and strain conversion vector B under each stress mode, and then substitute them into equation (1) to obtain the stiffness matrix under each stress mode. Step 2.4 Since the connecting rod (2) is a thin-walled box structure, the connecting rod (2) is modeled separately under shear deformation to obtain the pure shear stress mode. The matrix under the pure shear stress mode is linearly superimposed with the matrix under the bending stress mode to achieve correction. Then, the stiffness matrix under the corrected bending stress mode is integrated with the stiffness matrix under other stress modes to finally obtain the stiffness matrix K2 of the connecting rod (2). Step 2.5: Associate each parameter in the stiffness matrix K2 with the actual cross-sectional dimensions of the connecting rod (2); Step 3: Perform stiffness modeling on the front connection part (1) and the rear connection part (3) respectively, and use finite element software to extract the stiffness matrix K1 of the front connection part (1) and the stiffness matrix K3 of the rear connection part (3); Step 4: Transform K1, K2, and K3 to the same coordinate system and superimpose them to obtain the stiffness matrix of the entire robotic arm linkage. ; Step 5: Using the stiffness matrix Provide guidance on the design of robotic arm linkages.

2. The method for modeling the stiffness of a variable cross-section robotic arm link related to structural dimensions according to claim 1, characterized in that: In step 2.3, the strain transformation vector B for the tension / compression mode and the torsional mode is: (2)； The strain transformation vector B of the bending stress mode is: (3)； In equation (3) above, x and y are the x-axis and y-axis coordinates along the connecting rod (2), respectively. l Let N be the length of the connecting rod (2), and N be a shape function, where N is the force for tension / compression mode and torsion mode: （4）； The N value for the bending stress mode is: （5）； In equations (4) and (5) above: ； Where x is the absolute coordinate of the x-axis unit displacement field along the connecting rod (2); s is the relative coordinate of the x-axis unit displacement field along the connecting rod (2); The elastic matrix D under tension / compression and bending stress modes is as follows: （6）； In equation (6) above, E is Young's modulus; The elastic matrix D under torsional stress mode is: (7)； In the above equation (7), G is the shear modulus and μ is Poisson's ratio; Substituting equations (2)-(7) into equation (1) yields the stiffness matrix under each stress mode, where: The stiffness matrix under tension and compression modes is: （8）； In the above formula (8), A is the cross-sectional area of ​​the connecting rod part (2); l The length of the connecting rod (2); The stiffness matrix under torsional stress mode is: (9)； In the above formula (9), J is the torsional moment of inertia of the connecting rod (2); The stiffness matrix under bending stress mode is: (10)； In the above formula (10), The bending moment of inertia of the connecting rod (2) about the y-axis or about the z-axis.

3. The method for modeling the stiffness of a variable cross-section robotic arm link related to structural dimensions according to claim 2, characterized in that: In step 2.4, the parameters under the pure shear stress mode are as follows: Shape function N: (12）； Strain transformation vector B: (13)； Elasticity matrix D: （14）； In the above formula (14), For cross-sectional area, Let i be the cross-sectional shear coefficient, i = y, z; Substituting equations (13) and (14) into equation (1), we obtain the stiffness matrix for the pure shear mode: （15）； The stiffness matrix under the modified bending stress mode is obtained by superimposing the stiffness matrix (15) of the pure shear mode and the stiffness matrix (10) of the bending stress mode: （16）； The elements in equation (16) above are as follows: ； Integrating equations (16), (8), and (9) yields the final stiffness matrix of the connecting rod (2): （17）； The expressions for each element in equation (17) above are: ， 。 4. The method for modeling the stiffness of a variable cross-section robotic arm link related to structural dimensions according to claim 1 or 3, characterized in that: The parameters in the stiffness matrix K2 of the connecting rod (2) include the cross-sectional area of ​​the two end faces of the connecting rod (2), the moment of inertia of the cross-section of the two end faces of the connecting rod (2), the torsional moment of inertia of the two end faces of the connecting rod (2), and the shear coefficient of the cross-section of the two end faces of the connecting rod (2). In step 2.5, the parameters in K2 are related to the actual cross-sectional dimensions of the two end faces of the connecting rod (2) as follows: The cross-sectional areas of the two ends of the connecting rod (2) are: (18)； In the above formula (18), The width of the outer wall of the cross section. t is the length of the outer wall of the cross section, and t is the thickness of the four walls of the cross section. The moments of inertia of the two end faces of the connecting rod (2) are: （19）； （20）； The torsional moments of inertia of the two end faces of the connecting rod (2) are: （21）； The coefficients in equation (21) above Values ​​are selected by looking up a table; Shear coefficients of the two end faces of the connecting rod (2): （25）； In the above formula (25): The parameters are ; The parameters are ; and .

5. The method for modeling the stiffness of a variable cross-section robotic arm link related to structural dimensions according to claim 4, characterized in that: In step 2.5, the cross-sectional area of ​​equations (18)-(21) is... A、 Moment of inertia of cross section I、 Torsional moment of inertia of cross section J Perform the following equivalent processing: Cross-sectional area A : (22) ； Moment of inertia of cross section I : （23）； Torsional moment of inertia of cross section J : （24） 。 6. The method for modeling the stiffness of a variable cross-section robotic arm link in relation to structural dimensions according to claim 1, characterized in that: In step three, the front connection part (1) and the rear connection part (3) are both regarded as super elements with two nodes using finite element software. One end node is fixed and the stiffness matrix of the other end node is extracted to obtain the stiffness matrix K1 of the front connection part (1) and the stiffness matrix K3 of the rear connection part (3).

7. The method for modeling the stiffness of a variable cross-section robotic arm link related to structural dimensions according to claim 1, characterized in that: In step four, we first derive the coordinate transformation formula that transforms K1, K2, and K3 to the same coordinate system. T : (26)； In the above formula (26): ； in Local coordinate system representing the front connecting part (1), the connecting rod part (2), and the rear connecting part (3) For the global coordinate system The direction cosine; Then, transform each stiffness matrix to the same coordinate system: （27）； The stiffness matrix of the front connection part (1) in the global coordinate system is obtained according to the above equation (27). Stiffness matrix of link (2) Stiffness matrix of the rear connection part (3) ; Will , , Linear superposition yields the stiffness matrix of the entire robotic arm linkage. : （28）。

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