A method for calculating frequency response function of robot cutting system

Abstract

The present invention relates to the field of robotic cutting technology, resolving the technical issues faced by conventional methods for calculating frequency response functions, such as high cost, cumbersome operation, and limited adaptability to dynamic changes. The invention particularly relates to a method for calculating the frequency response function of a robotic cutting system, comprising the following steps: establishing a dynamic model of the robotic cutting system and its topological diagram based on a multi-body system transfer matrix method; deriving the complex-expanded transfer equations and complex-expanded transfer matrices for each element in the dynamic model; and deriving the complex-expanded total transfer equation and complex-expanded total transfer matrix of the robotic cutting system based on the dynamic model, its complex-expanded transfer equations and complex-expanded transfer matrix, and its topological diagram. The present invention can rapidly obtain the desired frequency response function without sacrificing accuracy, enabling real-time monitoring and immediate optimization of robotic cutting, and making dynamic characteristic analysis of robotic cutting more economical and efficient.

Description

Technical Field

[0001] The present invention relates to the technical field of robot cutting processing, and in particular to a method for calculating a frequency response function of a robot cutting processing system. Background Art

[0002] A robot's frequency response function (FRF) is a key parameter describing its dynamic characteristics. It reflects the robot's vibration behavior when subjected to external excitation. The FRF, which includes information such as the robot's structural natural frequency, damping ratio, and modal shape, serves as the foundation for analyzing and predicting the robot's vibration behavior and stability during machining.

[0003] However, traditional frequency response function calculation methods usually rely on offline experiments, such as impact hammer tests. These methods are not only time-consuming and labor-intensive, but also difficult to adapt to the dynamic changes of the robot in different working positions and postures. For example, the invention patent with publication number CN114789472A obtains the modal parameters of the robot at different rotation angles through a self-excitation method, and then synthesizes the frequency response function of the robot in the corresponding structural state based on modal theory and the modal expression of the frequency response function, as well as the calculated modal calibration factor. The invention patent with publication number CN116619357A uses the cutting force of the cutting process as the external excitation input, superimposes the modal responses of each order based on the modal superposition principle, calculates the physical space vibration displacement output, performs Fourier transform on the input force and output displacement respectively, and solves the tool tip frequency response function.

[0004] However, these methods rely on complex experimental setups and high-precision measurement equipment, and face challenges such as high cost, cumbersome operation, and limited adaptability to dynamic changes. In addition, the complexity of data processing also limits the widespread application of these methods. Summary of the Invention

[0005] In response to the shortcomings of the existing technology, the present invention provides a method for calculating the frequency response function of a robotic cutting machining system, which solves the technical problems faced by traditional methods of calculating frequency response functions, such as high cost, cumbersome operation, and limited adaptability to dynamic changes.

[0006] To solve the above technical problems, the present invention provides the following technical solution: a method for calculating the frequency response function of a robot cutting system, the method comprising the following steps:

[0007] S1. Establish the dynamic model and topology diagram of the robot cutting system based on the multi-body system transfer matrix method;

[0008] S2. Derive the complex extended transfer equations and complex extended transfer matrices of each element in the dynamic model;

[0009] S3. Based on the dynamic model, its complex extended transfer equation, complex extended transfer matrix and topological diagram, derive the complex extended total transfer equation and complex extended total transfer matrix of the robot cutting machining system;

[0010] S4. Based on the complex expansion total transfer equation and the complex expansion total transfer matrix, the relationship between the complex expansion state vector of the tool tip and the cutting force is derived;

[0011] S5. Calculate the frequency response function of the tool tip point of the robot cutting machining system based on the relationship between the complex extended state vector of the tool tip point and the cutting force.

[0012] Furthermore, in step S1, the specific process includes the following steps:

[0013] S11. Decompose the robot cutting system into several individual components and hinge components and uniformly number them according to the connection relationship between each robot body link, robot joint, end effector, and tool;

[0014] S12, using the fixed boundary and the free boundary of the robot cutting machining system as the input end and the output end of the dynamic model respectively;

[0015] S13. The connection relationships among the robot body connecting rod, robot joint, end effector and tool are modeled separately to obtain a dynamic model, specifically:

[0016] The robot's connecting rod is modeled as a spatial forced vibration rigid body. The robot's joints and the connection between the end effector and the tool are modeled as damped spatial elastic hinges. The tool can be equivalent to a spatial forced vibration rigid body or an Euler-Bernoulli beam, depending on its modeling purpose in different application scenarios.

[0017] S14. Establish a topological diagram based on the uniformly numbered individual components and hinge components.

[0018] Furthermore, in step S2, the specific process includes the following steps:

[0019] S21. Derive the complex expansion transfer equation and complex expansion transfer matrix of the spatial forced vibration rigid body. The derivation process is:

[0020] The state vectors of the input point I and the output point O in physical coordinates are defined as:

[0021]

[0022] Among them, x, y, z, θ x ,θ y ,θ z , m x 、m y 、m z,q x ,q y ,q z They represent linear displacement, angular displacement, internal moment and internal force in the inertial system respectively;

[0023] Solve using the complex exponential method, let

[0024]

[0025] in, is the complex amplitude of the state vector; λ is the complex eigenvalue, and the complex extended transfer equation of the spatial forced vibration rigid body is:

[0026]

[0027] in, is the complex extended transfer matrix, is the complex extended state vector;

[0028] Then, the complex expansion transfer matrix of the spatial forced vibration rigid body is for:

[0029]

[0030] in,

[0031]

[0032]

[0033] λ=iΩ

[0034] In the above formula, I3 is the third-order unit matrix; O i×j is an i-row, j-column, 0 matrix; m is the mass of the rigid body; are the coordinate antisymmetric matrices of the output point O, the center of mass point C, and the external force point D relative to the input point I in the rigid body coordinate system; J I is the inertia tensor matrix of the rigid body relative to the input point I; f is the expansion force matrix; F d (M d ) is the complex amplitude array of the concentrated external force / torque; Ω is the frequency of the external excitation force / torque; i is the imaginary unit;

[0035] S22. Derive the complex extended transfer equation and complex extended transfer matrix of the spatial forced vibration Euler-Bernoulli beam. The derivation process is as follows:

[0036] The equation defining the spatial forced vibration Euler-Bernoulli beam is:

[0037]

[0038] Using the complex exponential method to solve, the complex extended transfer equation of the spatial forced vibration Euler-Bernoulli beam is obtained as follows:

[0039]

[0040] Among them, the complex expanded transfer matrix of the spatial forced vibration Euler-Bernoulli beam is:

[0041]

[0042] in,

[0043]

[0044] in,

[0045] u 1,1 =u 10,10 =cosβ x l

[0046]

[0047] u 10,1 =β x EAsinβ x l

[0048]

[0049] u 6,2 =u 11,9 =λ y V(λ y l)

[0050]

[0051] u 5,3 =u 12,8 =-λ z V(λ z l)

[0052]

[0053] u 8,3 =u 12,5 =-EI y λ z 2 U(λ z l)

[0054]

[0055] u 2,2 =u 6,6 =u 9,9 =u11,11 =S(λ y l)

[0056] u 8,5 =EI y λ z V(λ z l)

[0057] u 9,6 =EI z λ y V(λ y l)

[0058] u 3,3 =u 5,5 =u 8,8 =u 12,12 =S(λ z l)

[0059]

[0060] λ=iΩ

[0061]

[0062]

[0063] In the above formula, l is the length of the beam; is the linear mass density; ρ is the density; E is the elastic modulus; G is the shear modulus; A is the cross-sectional area; I y is the moment of inertia of the cross section about the y-axis; I z is the moment of inertia of the cross section about the z axis; J p is the polar moment of inertia of the cross section about the x-axis; P(x1) and M(x1) are the complex amplitude arrays of the distributed external force and external moment, respectively;

[0064] X * (x1) and Θ * (x1) are differential equations and The corresponding special solution;

[0065] S23. Derive the complex expansion transfer equation and complex expansion transfer matrix of the damped space elastic hinge. The derivation process is:

[0066] According to the force balance of the damped space elastic hinge and Hooke's law, the complex expansion transfer equation of the damped space elastic hinge is:

[0067]

[0068] Among them, the complex expansion transfer matrix of the damped space elastic hinge is:

[0069]

[0070] in,

[0071]

[0072]

[0073] In the above formula, K x , K y , K z and C x 、C y 、C z They represent the linear stiffness and damping of the elastic hinge along the x, y, and z directions, respectively, K' x , K' y , K' z and C' x , C' y , C' z are the torsional stiffness and damping around the x, y, and z directions, respectively.

[0074] Furthermore, in step S3, the specific process includes:

[0075] According to the dynamic model and topological diagram of the rigid-flexible coupled robot cutting machining system, the complex extended total transfer equation of the n-degree-of-freedom series robot cutting machining system is established, namely:

[0076]

[0077] in, is the complex expanded total transfer matrix of the robot cutting machining system, which is obtained by multiplying the complex expanded transfer matrices of each element in sequence, that is:

[0078]

[0079] is the extended coordinate transformation matrix, which means the robot's i-th joint rotates around the j-axis by an angle of θ. It can be expressed as:

[0080]

[0081] Where,

[0082]

[0083] Furthermore, in step S4, the specific process includes the following steps:

[0084] S41. Calculate the complex extended total transfer matrix of the robot cutting system and write it in the form of a block matrix, that is:

[0085]

[0086] Where f is the expansion force matrix;

[0087] S42. Calculate the complex expanded total transfer equation of the robot cutting system to obtain:

[0088]

[0089] S43, will Written in the form of a block matrix, that is:

[0090]

[0091] S44. Substitute the boundary conditions into equation (13) and cross out The zero elements in The corresponding columns in Then the relationship between the complex expansion state vector of the cutting tool tip and the cutting force is:

[0092]

[0093] The boundary conditions are:

[0094]

[0095] In the above formula, is the cutting force of the cutting tool; is the expanded state vector of the cutting tool tip point.

[0096] Furthermore, in step S5, the specific process includes the following steps:

[0097] S51. Derive the frequency response function of the robot's cutting tool tip when the tool is modeled as a spatial forced vibration rigid body; the derivation process is:

[0098] From the relationship between the complex expansion state vector of the tool tip and the cutting force, it can be seen that the displacement complex amplitude of the cutting tool tip in the x direction is When the milling cutter is modeled as a rigid body subjected to forced vibration in space, the extended force matrix get

[0099]

[0100] Let F d =[F d,x ,0,0] T , M d =[0,0,0] T , the cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is:

[0101]

[0102] Where,

[0103]

[0104] λ=iΩ

[0105] Similarly, the displacement frequency response function of the cutting force acting in the y and z directions can be obtained. At this time, the displacement frequency response function matrix of the cutting tool tip is:

[0106]

[0107] By taking the derivative of the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity is obtained as follows:

[0108]

[0109] The frequency response function matrix of the cutting tool tip acceleration is:

[0110]

[0111] S52. Derive the frequency response function of the robot's cutting tool tip when the tool is modeled as a spatial forced vibration Euler-Bernoulli beam. The derivation process is as follows:

[0112] Let P(x1)=[P x ,P y ,P z ] T ,M(x1)=[M x ,M y ,M z ] T , we get:

[0113]

[0114] In the formula, K(u)=coshucosu, L(u)=coshusinu, M(u)=sinhucosu, N(u)=sinhusinu;

[0115] The cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is:

[0116]

[0117] Where,

[0118]

[0119] In the above formula, L m It is the length of action of the cutting distribution force on the tool during cutting;

[0120] Similarly, the frequency response function of the cutting tool tip displacement when the cutting force acts in the y and z directions can be obtained. At this time, the frequency response function matrix of the cutting tool tip displacement is:

[0121]

[0122] Where,

[0123] Similarly, the frequency response function matrix of the cutting tool tip velocity is H′(λ)=λH(λ), and the frequency response function matrix of the cutting tool tip acceleration is H″(λ)=λ 2 H(λ);

[0124] S53, calculating the frequency response function of the tool tip of the robot cutting system according to the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity, and the frequency response function matrix of the cutting tool tip acceleration;

[0125] S54. Calculate the damping ratio and natural frequency of the robot cutting system.

[0126] Furthermore, in step S53, the specific process includes:

[0127] The amplitude-frequency characteristic matrices |H(λ)|, |H′(λ)|, |H″(λ)| and the phase-frequency characteristic matrices arg(H(λ)), arg(H′(λ)), and arg(H″(λ)) of the displacement, velocity, and acceleration of the cutting tool tip are calculated respectively, which are the frequency response functions of the tool tip of the robot cutting machining system, where |·| is the modulus function and arg(·) is the argument function.

[0128] Furthermore, in step S54, the specific process includes the following steps:

[0129] S541, let the external excitation be 0, calculate the equation The solution is to find the complex eigenvalue λ of the robot cutting machining system;

[0130] S542. Calculate the damping ratio and natural frequency of the robot cutting system according to the complex eigenvalue λ, namely:

[0131]

[0132] Where ζ is the damping ratio and ω is the natural frequency of the undamped system.

[0133] By means of the above technical solution, the present invention provides a method for calculating the frequency response function of a robotic cutting system, which has at least the following beneficial effects:

[0134] 1. This invention can calculate the frequency response function of a robot in various postures from a complex modal perspective, providing a reference for stability analysis and optimization of the robot's machining process. It also simplifies the frequency response function calculation process, eliminating the need for high-precision measurement equipment and complex experimental setup, saving experimental preparation time and human resources, and reducing costs.

[0135] 2. The present invention can quickly obtain the required frequency response function without sacrificing accuracy, which makes it possible to monitor and optimize the robot cutting process in real time, making the dynamic characteristics analysis of the robot cutting process more economical and efficient.

[0136] 3. The present invention can avoid the experimental process and reduce the dependence on high-precision measuring equipment. It has good universality and real-time performance, can adapt to the dynamic characteristics analysis of the robot in various postures, and provide strong support for the stability analysis and optimization of the robot cutting process. BRIEF DESCRIPTION OF THE DRAWINGS

[0137] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0138] Figure 1 is a flow chart of the frequency response function calculation method of the present invention;

[0139] Figure 2 Schematic diagram of the dynamic model of the robot cutting processing system of the present invention;

[0140] Figure 3 Schematic diagram of the topology of the dynamic model of the robot cutting processing system of the present invention;

[0141] Figure 4 This is the amplitude-frequency characteristic diagram of the x-direction displacement of the cutting tool tip of the robot of the present invention;

[0142] Figure 5 This is the phase-frequency characteristic diagram of the x-direction displacement of the cutting tool tip of the robot of the present invention. DETAILED DESCRIPTION

[0143] To make the above-mentioned objectives, features, and advantages of the present invention more clearly understood, the present invention is further described below in detail with reference to the accompanying drawings and specific embodiments. This will enable a full understanding of how this application uses technical means to solve technical problems and achieve technical effects, and to implement the invention accordingly.

[0144] With the rapid development of the manufacturing industry, robotics is increasingly being used in industrial production. Robotic machining is particularly valued in fields such as aerospace, automotive manufacturing, and precision engineering for its high efficiency, precision, and flexibility. Robotic machining not only improves production efficiency but also ensures quality through precise control of machining parameters, thus meeting the machining needs of complex parts.

[0145] A robot's frequency response function (FRF) is a key parameter describing its dynamic characteristics. It reflects the robot's vibration behavior when subjected to external excitation. The FRF, which includes information such as the robot's structural natural frequency, damping ratio, and modal shape, serves as the foundation for analyzing and predicting the robot's vibration behavior and stability during machining.

[0146] However, traditional methods for calculating frequency response functions (FRFs) typically rely on offline experiments, such as impact hammer tests. These methods are not only time-consuming and labor-intensive, but also struggle to adapt to the dynamic changes of robots in different working positions and postures. Therefore, developing a new method that can rapidly calculate FRFs during actual machining processes is of great practical significance. This approach would not only provide a more accurate dynamic model for robotic machining but also provide a theoretical basis for the design and optimization of robotic control systems, thereby improving the performance of robotic machining.

[0147] In an article published in the academic journal Procedia Manufacturing, 2020, 154–158, Vinh Nguyen et al. first collected the force and vibration data of the robot during the cutting process through sensors, and then applied operational modal analysis technology to process the data. By analyzing the spectrum of the cutting force and the spectrum of the tool tip vibration, the frequency response function of the robot was estimated.

[0148] In an article published in the academic journal Experimental Techniques, 2023, 47:797–816, Mohammadi et al. used a desktop dynamometer to measure the input cutting force and an accelerometer to measure the vibration caused by it. The frequency response function was estimated by calculating the ratio of the cross power spectral density and the autopower spectral density of the force and vibration signals.

[0149] The invention patent with publication number CN114789472A obtains the modal parameters of the robot at different rotation angles through a self-excitation method, and then synthesizes the frequency response function of the robot in the corresponding structural state based on modal theory and the modal expression of the frequency response function, as well as the calculated modal calibration factor.

[0150] The invention patent with publication number CN116619357A uses the cutting force of the cutting process as the external excitation input, superimposes the modal responses of each order based on the modal superposition principle, calculates the physical space vibration displacement output, performs Fourier transform on the input force and output displacement respectively, and solves the tool tip frequency response function.

[0151] The above studies have proposed a variety of methods for calculating the frequency response functions of robotic cutting machining systems. However, these methods rely on complex experimental settings and high-precision measurement equipment, and face challenges such as high cost, cumbersome operation, and limited adaptability to dynamic changes. In addition, the complexity of data processing also limits the widespread application of these methods.

[0152] According to the technical defects of the existing technology, please refer to Figure 1-Figure 5 , shows a specific implementation of this embodiment. This embodiment calculates the frequency response function of the robot cutting system from the perspective of complex mode, simplifies the calculation process of the frequency response function, does not require high-precision measurement equipment and complex experimental settings, saves experimental preparation time and human resources, and reduces costs. Please refer to Figure 1 Taking the KUKA KR500 6-DOF serial robot machining system as an example, this embodiment proposes a method for calculating the frequency response function of a robotic cutting system. This method can quickly obtain the required frequency response function without sacrificing accuracy, enabling real-time monitoring and immediate optimization of robotic cutting processes, making dynamic characteristic analysis of robotic cutting processes more economical and efficient. The method includes the following steps:

[0153] S1. Based on the multi-body system transfer matrix method, the dynamic model and topology of the robot cutting processing system are established. The whole system is divided into units, that is, the dynamic model of the robot-end effector-tool rigid-flexible coupling is established. The established dynamic model and topology are as follows: Figure 2 、 Figure 3 As shown. As the optimal implementation method for implementing step S1, the specific process includes the following steps:

[0154] S11. Decompose the robot cutting system into several individual components and hinge components and uniformly number them according to the connection relationship between the robot's body links, robot joints, end effectors, and tools; Figure 2 As shown, the unified numbering is 0 to 14.

[0155] S12. The fixed boundary and free boundary of the robot cutting processing system are used as the input and output of the dynamic model respectively; among them, component No. 0 is the robot base, which is a fixed boundary and is regarded as the input of the robot dynamic model; component No. 14 is the cutting tool, which is a free boundary and is regarded as the output of the dynamic model.

[0156] S13. The connection relationships among the robot body connecting rod, robot joint, end effector and tool are modeled separately to obtain a dynamic model, specifically:

[0157] The robot body connecting rod is modeled as a space forced vibration rigid body; the robot joint and the connection between the end effector and the tool are modeled as damped space elastic hinges. The tool can be equivalent to a space forced vibration rigid body or Euler-Bernoulli beam depending on its different modeling purposes in different application scenarios; Figure 2 As shown, elements numbered 0, 2, 4, 6, 8, 10, and 12 are robot body links, which are modeled as spatial forced vibration rigid bodies; element No. 14 is a cutting tool, which can be equivalent to a spatial forced vibration rigid body or an Euler-Bernoulli beam depending on its different modeling purposes in different application scenarios. In this example, it is modeled as a spatial forced vibration rigid body; elements No. 1, 3, 5, 7, 9, and 11 are robot joints, and element No. 13 is the connection between the end effector and the tool, all of which are modeled as damped spatial elastic hinges; for the robot joints, considering the joint structure, its damped spatial elastic hinge is a torsion spring damper along the rotation axis and a spring damper orthogonal to this direction. The dynamic model of the entire robot cutting processing system is a tree structure, and its transmission direction is from element 0 to 14.

[0158] S14. Create a topological diagram based on the uniformly numbered individual components and hinge components, such as Figure 3 As shown, the topology diagram of the dynamic model is established according to the corresponding numbers of each body element and hinge element.

[0159] S2. Deriving the complex extended transfer equation and complex extended transfer matrix of each element in the dynamic model; as the optimal implementation method of step S2, the specific process includes the following steps:

[0160] S21. Derive the complex expansion transfer equation and complex expansion transfer matrix of the spatial forced vibration rigid body. The derivation process is:

[0161] The state vectors of the input point I and the output point O in physical coordinates are defined as:

[0162]

[0163] Among them, x, y, z, θ x ,θ y ,θ z , m x 、m y 、m z ,q x ,q y ,q z They represent linear displacement, angular displacement, internal moment and internal force in the inertial system respectively;

[0164] Solve using the complex exponential method, let

[0165]

[0166] in, is the complex amplitude of the state vector; λ is the complex eigenvalue, and the complex extended transfer equation of the spatial forced vibration rigid body is:

[0167]

[0168] in, is the complex extended transfer matrix, is the complex extended state vector;

[0169] Then, the complex expansion transfer matrix of the spatial forced vibration rigid body is for:

[0170]

[0171] in,

[0172]

[0173]

[0174] λ=iΩ

[0175] In the above formula, I3 is the third-order unit matrix; O i×j is an i-row, j-column, 0 matrix; m is the mass of the rigid body; are the coordinate antisymmetric matrices of the output point O, the center of mass point C, and the external force point D relative to the input point I in the rigid body coordinate system; J I is the inertia tensor matrix of the rigid body relative to the input point I; f is the expansion force matrix; F d (M d ) is the complex amplitude array of the concentrated external force (torque); Ω is the frequency of the external excitation force (torque); and i is the imaginary unit.

[0176] S22. Derive the complex extended transfer equation and complex extended transfer matrix of the spatial forced vibration Euler-Bernoulli beam. The derivation process is as follows:

[0177] The equation defining the spatial forced vibration Euler-Bernoulli beam is:

[0178]

[0179] Using the complex exponential method to solve, the complex extended transfer equation of the spatial forced vibration Euler-Bernoulli beam is obtained as follows:

[0180]

[0181] Among them, the complex expanded transfer matrix of the spatial forced vibration Euler-Bernoulli beam is:

[0182]

[0183] in,

[0184]

[0185]

[0186] in,

[0187] u 1,1 =u 10,10 =cosβ x l

[0188]

[0189] u 10,1 =β x EAsinβ x l

[0190]

[0191]

[0192] u 6,2 =u 11,9 =λ y V(λ y l)

[0193]

[0194] u 5,3 =u 12,8 =-λ z V(λ z l)

[0195]

[0196] u 8,3 =u 12,5 =-EI y λ z 2 U(λ z l)

[0197]

[0198] u 2,2 =u 6,6 =u 9,9 =u11,11 =S(λ y l)

[0199] u 8,5 =EI y λ z V(λ z l)

[0200] u 9,6 =EI z λ y V(λ y l)

[0201] u 3,3 =u 5,5 =u 8,8 =u 12,12 =S(λ z l)

[0202]

[0203] λ=iΩ

[0204]

[0205] In the above formula, l is the length of the beam; is the linear mass density; ρ is the density; E is the elastic modulus; G is the shear modulus; A is the cross-sectional area; I y is the moment of inertia of the cross section about the y-axis; I z is the moment of inertia of the cross section about the z axis; J p is the polar moment of inertia of the cross section about the x-axis; P(x1) and M(x1) are the complex amplitude arrays of the distributed external force and external moment, respectively;

[0206] X * (x1) and Θ * (x1) are differential equations and The corresponding special solution.

[0207] S23. Derive the complex expansion transfer equation and complex expansion transfer matrix of the damped space elastic hinge. The derivation process is:

[0208] According to the force balance of the damped space elastic hinge and Hooke's law, the complex expansion transfer equation of the damped space elastic hinge is:

[0209]

[0210] Among them, the complex expansion transfer matrix of the damped space elastic hinge is:

[0211]

[0212] in,

[0213]

[0214] In the above formula, K x , K y , K z and C x 、C y 、C z They represent the linear stiffness and damping of the elastic hinge along the x, y, and z directions, respectively, K' x , K' y , K' z and C' x , C' y , C' z are the torsional stiffness and damping around the x, y, and z directions, respectively.

[0215] S3. Based on the dynamic model, its complex extended transfer equation, complex extended transfer matrix, and topological diagram, derive the complex extended total transfer equation and complex extended total transfer matrix of the robot cutting machining system. As the optimal implementation method of step S3, the specific process includes:

[0216] According to the dynamic model and topological diagram of the rigid-flexible coupled robot cutting machining system, the complex extended total transfer equation of the n-degree-of-freedom series robot cutting machining system is established, namely:

[0217]

[0218] in, is the complex expanded total transfer matrix of the robot cutting machining system, which is obtained by multiplying the complex expanded transfer matrices of each element in sequence, that is:

[0219]

[0220] is the extended coordinate transformation matrix, which means the robot's i-th joint rotates around the j-axis by an angle of θ. It can be expressed as:

[0221]

[0222] Where,

[0223]

[0224] Among them, taking the KUKA KR500 6-DOF serial robot cutting machining system as an example, the complex expanded total transfer equation of the robot cutting machining system is established based on the dynamic model of the rigid-flexible coupling multi-body system and its corresponding topological graph:

[0225]

[0226] Among them, the complex expanded total transfer matrix of the robot cutting machining system is:

[0227]

[0228] For rigid body elements b = 2, 4, 6, 8, 10, and 12, the complex extended transfer matrix for:

[0229]

[0230] Where,

[0231]

[0232] Among them, I3 is the third-order unit matrix, O i×j is a matrix with row i and column j, and m b is the mass of the b-th rigid body, are the output point O in the coordinate system of the bth individual component b , center of mass point C b , external force application point D b Relative to input point I b The coordinate antisymmetric matrix, J b,I is the rigid body relative to the input point I b The inertia tensor matrix, f is the expansion force matrix, F d (M d ) is the complex amplitude array of concentrated external force (torque), O b =(o b1 ,o b2 ,o b3 ) is the output point coordinate, C b =(c b1 ,c b2 ,c b3 ) is the coordinate of the center of mass, D b =(d b1 ,d b2 ,d b3 ) is the coordinate of the external force point, λ is the complex eigenvalue;

[0233] For hinge elements j = 1, 3, 5, 7, 9, 11, and 13, the complex expanded transfer matrix is for

[0234]

[0235] Where,

[0236]

[0237] in, represents the multilinear stiffness coefficient matrix of the hinge element numbered j, represents the complex torsional stiffness coefficient matrix of the hinge element numbered j, K x , K y , K z and C x 、C y 、C z They represent the linear stiffness and damping of the elastic hinge along the x, y, and z directions, respectively, K' x , K' y , K' z and C' x , C' y , C' z are the torsional stiffness and damping around the x, y, and z directions, respectively.

[0238] For component No. 14, it is modeled as a space forced vibration rigid body. According to the principle of equivalent force system, the cutting distributed force (moment) can be equivalent to the cutting concentrated force (moment), and the cutting force action point can be determined. At this time, the complex expansion transfer matrix of the No. 14 cutting tool is the complex expansion transfer matrix of the space forced vibration rigid body, where F d (M d ) is the complex amplitude of the cutting concentrated force (moment).

[0239] S4. Based on the complex-expanded total transfer equation and the complex-expanded total transfer matrix, a relationship between the complex-expanded state vector of the tool tip and the cutting force is derived. As an optimal method for implementing step S4, the specific process includes the following steps:

[0240] S41. Calculate the complex extended total transfer matrix of the robot cutting system and write it in the form of a block matrix, that is:

[0241]

[0242] Right now:

[0243]

[0244] Where f is the expansion force matrix;

[0245] S42. Calculate the complex expanded total transfer equation of the robot cutting machining system and obtain:

[0246]

[0247] Right now:

[0248]

[0249] S43, will Written in the form of a block matrix, that is:

[0250]

[0251] Right now:

[0252]

[0253] S44. Substitute the boundary conditions into equation (13) and cross out The zero elements in The corresponding columns in Then the relationship between the complex expansion state vector of the cutting tool tip and the cutting force is:

[0254]

[0255] The boundary conditions are:

[0256]

[0257] Right now:

[0258]

[0259] In the above formula, is the cutting force of the cutting tool; is the expanded state vector of the cutting tool tip point.

[0260] S5. Calculate the frequency response function of the tool tip of the robot cutting system based on the relationship between the complex extended state vector of the tool tip and the cutting force. As an optimal implementation method for implementing step S5, the specific process includes the following steps:

[0261] S51. Derive the frequency response function of the robot's cutting tool tip when the tool is modeled as a spatial forced vibration rigid body; the derivation process is:

[0262] From the relationship between the complex expansion state vector of the tool tip and the cutting force, it can be seen that the displacement complex amplitude of the cutting tool tip in the x direction is When the milling cutter is modeled as a rigid body subjected to forced vibration in space, the extended force matrix get

[0263]

[0264] Let F d =[F d,x ,0,0] T , M d =[0,0,0] T , the cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is:

[0265]

[0266] Where,

[0267]

[0268] λ=iΩ

[0269] Similarly, the frequency response function of the cutting tool tip displacement when the cutting force acts in the y and z directions can be obtained. At this time, the frequency response function matrix of the cutting tool tip displacement is:

[0270]

[0271] By taking the derivative of the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity is obtained as follows:

[0272]

[0273] The frequency response function matrix of the cutting tool tip acceleration is:

[0274]

[0275] S52. Derive the frequency response function of the robot's cutting tool tip when the tool is modeled as a spatial forced vibration Euler-Bernoulli beam. The derivation process is as follows:

[0276] Let P(x1)=[P x ,P y ,P z ] T ,M(x1)=[M x ,M y ,M z ] T , we get:

[0277]

[0278] In the formula, K(u)=coshucosu, L(u)=coshusinu, M(u)=sinhucosu, N(u)=sinhusinu;

[0279] The cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is:

[0280]

[0281] Where,

[0282]

[0283] In the above formula, L m It is the length of action of the cutting distribution force on the tool during cutting;

[0284] Similarly, the frequency response function of the cutting tool tip displacement when the cutting force acts in the y and z directions can be obtained. At this time, the frequency response function matrix of the cutting tool tip displacement is:

[0285]

[0286] Where,

[0287] Similarly, the frequency response function matrix of the cutting tool tip velocity is H′(λ)=λH(λ), and the frequency response function matrix of the cutting tool tip acceleration is H″(λ)=λ 2 H(λ).

[0288] S53. Calculate the frequency response function of the tool tip of the robot cutting system based on the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity, and the frequency response function matrix of the cutting tool tip acceleration, that is:

[0289] The amplitude-frequency characteristic matrices |H(λ)|, |H′(λ)|, |H″(λ)| and the phase-frequency characteristic matrices arg(H(λ)), arg(H′(λ)), and arg(H″(λ)) of the displacement, velocity, and acceleration of the cutting tool tip are calculated respectively, which are the frequency response functions of the tool tip of the robot cutting machining system, where |·| is the modulus function and arg(·) is the argument function.

[0290] S54, calculating the damping ratio and natural frequency of the robot cutting processing system, as the optimal implementation method of step S54, the specific process includes the following steps:

[0291] S541, let the external excitation be 0, calculate the equation The solution is to find the complex eigenvalue λ of the robot cutting machining system;

[0292] S542. Calculate the damping ratio and natural frequency of the robot cutting system according to the complex eigenvalue λ, namely:

[0293]

[0294] Where ζ is the damping ratio and ω is the natural frequency of the undamped system.

[0295] Specifically, take the KUKA KR500 6-DOF serial robot cutting processing system as an example, as follows:

[0296] First, the frequency response function of the robot cutting tool tip is derived. From Equation (14), we can see that the displacement complex amplitude of the cutting tool tip in the x direction is When the milling cutter is modeled as a rigid body with forced vibration in space:

[0297] By the expansion force matrix have to

[0298] Let F d =[F d,x,0,0] T , M d =[0,0,0] T , the cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is:

[0299]

[0300] Where, λ=iΩ, Ω is the cutting force frequency;

[0301] Similarly, the frequency response function of the cutting tool tip displacement when the cutting force acts in the y and z directions can be obtained. At this time, the frequency response function matrix of the cutting tool tip displacement is:

[0302]

[0303] By taking the derivative of the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity is obtained as follows:

[0304]

[0305] The frequency response function matrix of the cutting tool tip acceleration is:

[0306]

[0307] Secondly, the frequency response function of the cutting tool tip of the robot machining system is calculated. The displacement, velocity, and acceleration amplitude-frequency characteristic matrices |H(λ)|, |H′(λ)|, and |H″(λ)| and the displacement, velocity, and acceleration phase-frequency characteristic matrices arg(H(λ)), arg(H′(λ)), and arg(H″(λ)) are calculated respectively, where |·| is the modulus function and arg(·) is the argument function. In this example, only the displacement amplitude-frequency characteristic and phase-frequency characteristic of the cutting tool tip in the x-direction are calculated.

[0308] The amplitude-frequency characteristic diagram and phase-frequency characteristic diagram of the x-direction displacement of the cutting tool tip point of the KUKA KR500 robot machining system drawn by the present invention are shown as follows: Figure 4 、 Figure 5 shown.

[0309] Based on the known robot dynamic parameters, the present invention directly utilizes the multi-body system transfer matrix complex modal method to efficiently calculate the frequency response function. By constructing a multi-body dynamic model of the robot-end effector-tool machining system, determining the system dynamic model topology, and deducing the complex extended transfer matrix of each robot component from the perspective of complex modality, the system complex extended total transfer equation is established, and the cutting tool tip frequency response function is calculated. This invention can avoid the experimental process and reduce the dependence on high-precision measurement equipment. It has good universality and real-time performance, can adapt to the dynamic characteristics analysis of the robot in various positions, and provide strong support for the stability analysis and optimization of the robot cutting process.

[0310] Those skilled in the art will appreciate that all or part of the steps in the above-mentioned embodiment methods can be accomplished by instructing the relevant hardware through a program. Therefore, the present application may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Furthermore, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0311] Each embodiment in this specification is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to in detail. For the above embodiments, since they are basically similar to the method embodiments, the description is relatively simple. For relevant parts, please refer to the partial description of the method embodiments.

[0312] The above embodiments provide a detailed introduction to the present invention. Specific examples are used herein to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core ideas. At the same time, for those skilled in the art, according to the ideas of the present invention, there may be changes in the specific implementation methods and application scopes. In summary, the contents of this specification should not be understood as limiting the present invention.

Claims

1. A method for calculating the frequency response function of a robot cutting system, characterized in that: The method comprises the following steps: S1. Establish the dynamic model and topology diagram of the robot cutting system based on the multi-body system transfer matrix method; S2. Derive the complex extended transfer equations and complex extended transfer matrices of each element in the dynamic model; S3. Based on the dynamic model, its complex extended transfer equation and complex extended transfer matrix, and the topological diagram, derive the complex extended total transfer equation and complex extended total transfer matrix of the robot cutting machining system; S4. Based on the complex expansion total transfer equation and the complex expansion total transfer matrix, the relationship between the complex expansion state vector of the tool tip and the cutting force is derived; S5. Calculate the frequency response function of the tool tip of the robot cutting system based on the relationship between the complex extended state vector of the tool tip and the cutting force. The specific process includes the following steps: S51. Derive the frequency response function of the robot's cutting tool tip when the tool is modeled as a spatial forced vibration rigid body; the derivation process is: From the relationship between the complex expansion state vector of the tool tip and the cutting force, it can be seen that the displacement complex amplitude of the cutting tool tip in the x direction is When the milling cutter is modeled as a rigid body subjected to forced vibration in space, the extended force matrix get Let F d =[F d,x ,0,0] T , M d =[0,0,0] T , the cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is: Where, λ=iΩ Similarly, the displacement frequency response function of the cutting force acting in the y and z directions can be obtained. At this time, the displacement frequency response function matrix of the cutting tool tip is: By taking the derivative of the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity is obtained as follows: The frequency response function matrix of the cutting tool tip acceleration is: S52. Derive the frequency response function of the robot's cutting tool tip when the tool is modeled as a spatial forced vibration Euler-Bernoulli beam. The derivation process is as follows: Let P(x1)=[P x ,P y ,P z ] T ,M(x1)=[M x ,M y ,M z ] T , we get: In the formula, K(u)=coshucosu, L(u)=coshusinu, M(u)=sinhucosu, N(u)=sinhusinu; The cutting force acts in the x-direction, and the displacement frequency response function of the tool tip in the x-direction is: Where, In the above formula, L m It is the length of action of the cutting distribution force on the tool during cutting; Similarly, the frequency response function of the cutting tool tip displacement when the cutting force acts in the y and z directions can be obtained. At this time, the frequency response function matrix of the cutting tool tip displacement is: Where, Similarly, the frequency response function matrix of the cutting tool tip velocity is H′(λ)=λH(λ), and the frequency response function matrix of the cutting tool tip acceleration is H″(λ)=λ 2 H(λ); S53, calculating the frequency response function of the tool tip of the robot cutting system according to the frequency response function matrix of the cutting tool tip displacement, the frequency response function matrix of the cutting tool tip velocity, and the frequency response function matrix of the cutting tool tip acceleration; S54. Calculate the damping ratio and natural frequency of the robot cutting system.

2. The frequency response function calculation method according to claim 1, characterized in that: In step S1, the specific process includes the following steps: S11. Decompose the robot cutting system into several individual components and hinge components and uniformly number them according to the connection relationship between each robot body link, robot joint, end effector, and tool; S12, using the fixed boundary and the free boundary of the robot cutting machining system as the input end and the output end of the dynamic model respectively; S13. The connection relationships among the robot body connecting rod, robot joint, end effector and tool are modeled separately to obtain a dynamic model, specifically: The robot's connecting rod is modeled as a spatial forced vibration rigid body. The robot's joints and the connection between the end effector and the tool are modeled as damped spatial elastic hinges. The tool can be equivalent to a spatial forced vibration rigid body or an Euler-Bernoulli beam, depending on its modeling purpose in different application scenarios. S14. Establish a topological diagram based on the uniformly numbered individual components and hinge components.

3. The frequency response function calculation method according to claim 1, characterized in that: In step S2, the specific process includes the following steps: S21. Derive the complex expansion transfer equation and complex expansion transfer matrix of the spatial forced vibration rigid body. The derivation process is: The state vectors of the input point I and the output point O in physical coordinates are defined as: Among them, x, y, z, θ x ,θ y ,θ z , m x 、m y 、m z ,q x ,q y ,q z They represent linear displacement, angular displacement, internal moment and internal force in the inertial system respectively; Solve using the complex exponential method, let in, is the complex amplitude of the state vector; λ is the complex eigenvalue, and the complex extended transfer equation of the spatial forced vibration rigid body is: in, is the complex extended transfer matrix, is the complex extended state vector; Then, the complex expansion transfer matrix of the spatial forced vibration rigid body is for: in, λ=iΩ In the above formula, I3 is the third-order unit matrix; O i×j is an i-row, j-column, 0 matrix; m is the mass of the rigid body; are the coordinate antisymmetric matrices of the output point O, the center of mass point C, and the external force point D relative to the input point I in the rigid body coordinate system; J I is the inertia tensor matrix of the rigid body relative to the input point I; f is the expansion force matrix; F d (M d ) is the complex amplitude array of the concentrated external force / torque; Ω is the frequency of the external excitation force / torque; i is the imaginary unit; S22. Derive the complex extended transfer equation and complex extended transfer matrix of the spatial forced vibration Euler-Bernoulli beam. The derivation process is as follows: The equation defining the spatial forced vibration Euler-Bernoulli beam is: Using the complex exponential method to solve, the complex extended transfer equation of the spatial forced vibration Euler-Bernoulli beam is obtained as follows: Among them, the complex expanded transfer matrix of the spatial forced vibration Euler-Bernoulli beam is: in, in, and 1,1 =and 10,10 =cosβ x the you 10,1 =b x EAsinβ x l you 6,2 =u 11,9 =λ y V(λ y l) you 5,3 =u 12,8 =-λ z V(λ z l) about 8,3 =u 12,5 =-NO y λ z 2 U(λ z l) in 2,2 =in 6,6 =in 9,9 =in 11,11 =S(λ y l) you 8,5 =NO y λ z V(λ z l) you 9,6 =NO z λ y V(λ y l) in 3,3 =in 5,5 =in 8,8 =in 12,12 =S(λ z l) λ=iΩ In the above formula, l is the length of the beam; is the linear mass density; ρ is the density; E is the elastic modulus; G is the shear modulus; A is the cross-sectional area; I y is the moment of inertia of the cross section about the y-axis; I z is the moment of inertia of the cross section about the z axis; J p is the polar moment of inertia of the cross section about the x-axis; P(x1) and M(x1) are the complex amplitude arrays of the distributed external force and external moment, respectively; X * (x1) and Θ * (x1) are differential equations and The corresponding special solution; S23. Derive the complex expansion transfer equation and complex expansion transfer matrix of the damped space elastic hinge. The derivation process is: According to the force balance of the damped space elastic hinge and Hooke's law, the complex expansion transfer equation of the damped space elastic hinge is: Among them, the complex expansion transfer matrix of the damped space elastic hinge is: in, In the above formula, K x , K y , K z and C x 、C y 、C z They represent the linear stiffness and damping of the elastic hinge along the x, y, and z directions, respectively, K' x , K' y , K' z and C' x , C' y , C' z are the torsional stiffness and damping around the x, y, and z directions, respectively.

4. The frequency response function calculation method according to claim 1, characterized in that: In step S3, the specific process includes: According to the dynamic model and topological diagram of the rigid-flexible coupled robot cutting machining system, the complex extended total transfer equation of the n-degree-of-freedom series robot cutting machining system is established, namely: in, is the complex expanded total transfer matrix of the robot cutting machining system, which is obtained by multiplying the complex expanded transfer matrices of each element in sequence, that is: is the extended coordinate transformation matrix, which means the robot's i-th joint rotates around the j-axis by an angle of θ. It can be expressed as: Where, 5. The frequency response function calculation method according to claim 1, characterized in that: In step S4, the specific process includes the following steps: S41. Calculate the complex extended total transfer matrix of the robot cutting system and write it in the form of a block matrix, that is: Where f is the expansion force matrix; S42. Calculate the complex expanded total transfer equation of the robot cutting system to obtain: S43, will Written in the form of a block matrix, that is: S44. Substitute the boundary conditions into equation (13) and cross out The zero elements in The corresponding columns in Then the relationship between the complex expansion state vector of the cutting tool tip and the cutting force is: The boundary conditions are: In the above formula, is the cutting force of the cutting tool; is the expanded state vector of the cutting tool tip point.

6. The frequency response function calculation method according to claim 1, characterized in that: In step S53, the specific process includes: The amplitude-frequency characteristic matrices |H(λ)|, |H′(λ)|, |H″(λ)| and the phase-frequency characteristic matrices arg(H(λ)), arg(H′(λ)), and arg(H″(λ)) of the displacement, velocity, and acceleration of the cutting tool tip are calculated respectively, which are the frequency response functions of the tool tip of the robot cutting machining system, where |·| is the modulus function and arg(g) is the argument function.

7. The frequency response function calculation method according to claim 1, characterized in that: In step S54, the specific process includes the following steps: S541, let the external excitation be 0, calculate the equation The solution is to find the complex eigenvalue λ of the robot cutting machining system; S542. Calculate the damping ratio and natural frequency of the robot cutting system according to the complex eigenvalue λ, namely: Where ζ is the damping ratio and ω is the natural frequency of the undamped system.

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