A Follow-up Hybrid Support Structure for a Milling Robot, Vibration Modeling and Its Control Method

Abstract

The present invention discloses a follow-up hybrid support structure for a milling robot, vibration modeling and its control method, which relates to the field of milling. The follow-up hybrid support structure for the milling robot is a three-stage hybrid configuration of "parallel - series - parallel", including a support point equivalent unit, a robot equivalent unit, and a workpiece supported area equivalent unit. The support point equivalent units are first connected in parallel, and the parallel-connected support point equivalent units are then connected in series with the robot equivalent unit. The series-connected support point equivalent units, robot equivalent units are then connected in parallel with the workpiece supported area equivalent system. The vibration modeling includes the vibration equation corresponding to the vibration modeling, and its control method includes two steps: control equation decoupling and design of constraint following control. The follow-up hybrid support structure for the milling robot and its constraint following control method adopting the robot follow-up support strategy can achieve the servo requirements of vibration control with high precision and low cost.

Description

Technical Field

[0001] The present invention relates to the field of milling, and in particular to a follow-up hybrid support structure for a milling robot, vibration modeling, and its control method. Background Art

[0002] Thin-walled parts are widely used in the fields of aviation and aerospace. However, their weak rigidity leads to easy deformation and tremors during the machining process, further affecting the machining accuracy and surface quality. Applying a suitable support strategy can effectively solve this problem. The three mainstream support strategies are the profiling support strategy, the multi-point discrete support strategy, and the follow-up support strategy. Among them, the profiling support strategy can provide a stable support effect, but it is expensive and has poor compatibility. In contrast, the multi-point discrete support strategy can adapt to components with different geometric shapes by adjusting the height and angle of each support node. However, there are gaps between the nodes, making it difficult to ensure the local stiffness and machining accuracy at the gaps. The follow-up support strategy can provide a low-cost and highly flexible support effect for the machining of thin-walled workpieces, effectively avoiding the above problems. This strategy can ensure sufficient local stiffness at the vibration excitation source at all times while saving costs by driving the support head to move synchronously with the machining tool.

[0003] Patent CN110332277B proposes a vibration control device for curved thin-walled parts based on permanent magnet drive, which can use one drive source to achieve support at multiple positions. However, there are large gaps between the support nodes, and it is difficult to ensure the local stiffness compared with the follow-up support of a robot.

[0004] Patent CN114248134B proposes a follow-up support vibration suppression adjustment method suitable for the machining of thin-walled parts with different thicknesses. Its vibration suppression mechanism includes an active side device and a driven side device, and vibration suppression is achieved through the active side telescopic device according to the requirements of different thicknesses and milling depths. It considers the change in the wall thickness of the workpiece, but does not consider the change in stiffness caused by the pose change of the actuator.

[0005] To describe and predict the deformation and vibration of the workpiece under the follow-up support strategy, it is necessary to establish a dynamic model of the follow-up support strategy, which also provides a theoretical basis for the subsequent optimization design of the support effect. The existing research on the follow-up support strategy has not fully considered the coupled vibration and the relationship between stiffness hybridization of the three heterogeneous systems of the workpiece, the support head, and the robot.

[0006] Therefore, there is an urgent need for a follow-up hybrid support structure for a milling robot, vibration modeling, and its control method that consider the relationship between coupled vibration and stiffness hybridization to achieve the servo demand of vibration control with high precision and low cost. Summary of the Invention

[0007] To solve the above problems, in view of the complex coupling and time-varying dynamic characteristics of the follow-up support strategy, this application proposes a set of "parallel - series - parallel" three-level heterogeneous hybrid configuration and its vibration modeling and control method.

[0008] Among them, a follow-up hybrid support structure for a milling robot, the follow-up hybrid support structure for the milling robot is a "parallel - series - parallel" three-level hybrid configuration, including a support point equivalent unit, a robot equivalent unit, and a workpiece supported area equivalent unit;

[0009] The support point equivalent units are first connected in parallel, and the parallel-connected support point equivalent units are then connected in series with the robot equivalent unit. The series-connected support point equivalent units, robot equivalent units, and the workpiece supported area equivalent system are then connected in parallel;

[0010] The support point equivalent unit includes a support node, and the support node is connected by a stud between a needle-shaped cylinder and a universal ball. The needle-shaped cylinder at one end of the support node is arranged on the flange surface of the robot equivalent unit, forming a mass bundling with the robot, and the universal ball at the other end contacts the workpiece surface, forming a mass bundling with the workpiece;

[0011] The needle-shaped cylinder inside the support node is provided with gas.

[0012] Vibration modeling of a follow-up hybrid support structure for a milling robot, the vibration equation corresponding to the vibration modeling is:

[0013] ;

[0014] ;

[0015] ;

[0016] ;

[0017] Among them, is time, is the mass matrix of the vibration equation, is the stiffness matrix of the vibration equation, is the force vector of the vibration equation, is the displacement vector of the vibration equation, is the velocity vector of the vibration equation, is the acceleration vector of the vibration equation, is the axial milling acting force, is the total support force generated by air pressure, and are respectively the displacements of the workpiece supported area and the robot flange along the normal direction of the workpiece, is the equivalent mass and equivalent stiffness of the supported area of the workpiece, is the equivalent mass and equivalent stiffness of the robot, is the number of support points of the multi-point support head, is the equivalent stiffness of a single support point, is the mass of the bundled part of the support node and the workpiece, is the mass of the bundled part of the support node and the robot, is the damping matrix for the equivalent vibration equation, is the equivalent proportional coefficient of the mass matrix, is the equivalent proportional coefficient of the stiffness matrix.

[0018] Preferably, the expression for the equivalent stiffness of the supported area of the workpiece is:

[0019] ;

[0020] where, is the abscissa of the point in the supported area of the workpiece, is the ordinate of the point in the supported area of the workpiece, is the flexural stiffness of the workpiece, is half of the length of the workpiece, is half of the width of the workpiece.

[0021] Preferably, for a serial robot with degrees of freedom, the generalized coordinates are where is the rotation angle of the th joint of the robot, and the equivalent stiffness matrix in the normal direction of the robot is:

[0022] ;

[0023] where, is the rotation matrix between the base coordinate system and the end coordinate system of the robot, is the dimensional linear velocity Jacobian matrix of the robot, is the joint stiffness matrix,

[0024] Preferably, the expression for the equivalent stiffness of a single support point is:

[0025] ;

[0026] where, is the air pressure in the rodless cavity of the support point cylinder, is the adiabatic coefficient of the ideal gas, is the cross-sectional area of the rodless cavity of the cylinder, is the preset distance for the piston rod of the needle-type cylinder to extend.

[0027] A control method for vibration modeling of a milling robot's follow-up hybrid support structure includes two steps: decoupling of control equations and design constraint following control.

[0028] Preferably, the specific content of decoupling the control equations is as follows:

[0029] Decouple the vibration equation corresponding to the active control term to obtain the equivalent control equation:

[0030] ;

[0031] In the formula:

[0032] ;

[0033] and are two state variables to be controlled, then define as the input matrix, i.e., the input vector, of the milling robot's follow-up hybrid support structure, are respectively the damping matrix, stiffness matrix, and force vector in the equivalent control equation;

[0034] The control task is to design such that meets the requirements.

[0035] Preferably, it includes:

[0036] Write the control objective in the form of servo constraints:

[0037] ;

[0038] In the formula, , is the target actively designed by the controller to control the vibration amplitude to reduce vibration energy or adjust the vibration frequency to avoid resonance according to specific cases;

[0039] Arrange the control objective into matrix form:

[0040] ;

[0041] In the formula:

[0042] ;

[0043] Take the time derivative of both sides of the matrix form of the control objective and rewrite it as a second-order form of constraint:

[0044] ;

[0045] wherein:

[0046] ;

[0047] For all , there is ;

[0048] For the given constant matrix simultaneously , there exists a constant For all , ;

[0049] Define the constraint following error: ;

[0050] Barometric pressure expression: ;

[0051] wherein: ;

[0052] ;

[0053] In the formula, is the control parameter, is the nominal control term, is the generalization of the Udwadia-Kalaba principle in the underactuated system, which has the minimum norm among all the control forces that satisfy the system servo motion requirements; is the feedback control term, which is used to compensate for the deviation between the actual state and the desired state of the system;

[0054] The constraint following error satisfies: for any positive real number , there exists a finite positive real number , if , then for all , there is , .

[0055] In summary, for the follow-up hybrid support structure, vibration modeling and control method of a milling robot according to the present invention, compared with the traditional technology, the present invention proposes a three-stage hybrid follow-up support vibration configuration of "parallel - series - parallel". This configuration fully considers the time-varying stiffness field characteristics of the three heterogeneous systems of the workpiece, the support head and the robot and gives the analytical models of each stiffness. A control method for vibration modeling of the follow-up hybrid support structure of a milling robot is proposed. This method is an active vibration control strategy for the follow-up support system based on the constraint following theory. This strategy decouples the controllable terms of the system to obtain an equivalent underactuated control model. At the same time, it is proved from the Lyapunov stability and verified by comparative simulation that the algorithm can meet the vibration control objectives. The proposed control method can achieve the servo requirements of vibration control with high precision and low cost.

[0056] The technical method of the present invention will be further described in detail below with reference to the drawings and embodiments. Description of the Drawings

[0057] Figure 1 Figures are diagrams of three mainstream support strategies for thin-walled parts;

[0058] Figure 2 Figure is a schematic diagram of the three-stage hybrid configuration of the follow-up support domain of the present invention;

[0059] Figure 3 Distribution of the equivalent stiffness of the workpiece at different positions of the workpiece;

[0060] Figure 4 Relationship between the equivalent stiffness of the support point, the extension distance of the cylinder and the input air pressure;

[0061] Figure 5 Distribution of the normal equivalent stiffness of the support point within the working space;

[0062] Figure 6 Vibration displacement of the workpiece under different control methods;

[0063] Figure 7 Comparison of the total error of the vibration displacement of the workpiece under different control methods;

[0064] Figure 8 Air pressure under different control methods;

[0065] Figure 9 Comparison of the total cost of air pressure under different control methods.

[0066] Reference Signs

[0067] 1. Support node; 11. Needle cylinder; 12. Universal ball; 2. Workpiece; 3. Gas; 4. Flange. Detailed Embodiment

[0068] The technical method of the present invention will be further described below with reference to the accompanying drawings and embodiments. It should be noted that: unless otherwise specifically stated, the relative arrangements of components and steps, numerical expressions and values set forth in these embodiments do not limit the scope of the present application.

[0069] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way a limitation on the present application, its application, or use.

[0070] Technologies, systems, and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but where appropriate, the technologies, systems, and devices should be regarded as part of the specification.

[0071] In all examples shown and discussed herein, any specific values should be construed as merely exemplary and not as a limitation. Thus, other examples of the exemplary embodiments may have different values.

[0072] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meaning as understood by those of ordinary skill in the field to which the present invention pertains.

[0073] As Figure 1 shown, the three mainstream support strategies are the profiling support strategy, the multi-point discrete support strategy, and the follow-up support strategy. The above three support strategies have problems such as high cost, poor compatibility, difficulty in ensuring the local stiffness and machining accuracy at the gaps, and failure to fully consider the coupled vibration and stiffness series-parallel relationship among the workpiece, the support head, and the robot, which are three heterogeneous systems.

[0074] The present invention provides a follow-up series-parallel support structure for a milling robot, vibration modeling, and its control method. First, a second-order vibration model of the follow-up support strategy is constructed, and the modeling is simplified by mass bundling while capturing the main stiffness characteristics of the system. Second, time-varying stiffness analytical expressions of the three main stiffnesses, namely the local area of the workpiece, the gas spring at the support point, and the normal direction of the robot end, are established respectively through the elastic deformation equation, the adiabatic gas state equation, and the robot force-Jacobian mapping, facilitating real-time substitution calculation in the control process. Finally, the controllable air pressure term is decoupled from both sides of the vibration equation to obtain an equivalent underactuated system, and a constrained following control algorithm is designed. Comparative simulations show that the proposed control method can achieve the servo requirements of vibration control with high precision and low cost.

[0075] Embodiment 1

[0076] To perform certain simplified modeling and capture the main vibration characteristics of the system, the present invention does not focus on the vibration conditions at all positions on the entire workpiece surface, but instead focuses on vibration modeling within the support area that moves following the vibration excitation source. At the same time, since the thickness dimension of the thin-walled part is much smaller than the other two dimensions, this results in the weakest normal stiffness of the workpiece, and the resulting deformation and vibration are also more intense. Therefore, the modeling of the present invention focuses on the vibration along the normal direction of the workpiece.

[0077] Hypothesis 1: The vibration conditions of the workpiece within the support area are consistent at each moment.

[0078] Since the size of the support area is very small relative to the entire thin-walled component, it can be approximately considered that the physical parameters and vibration conditions of the workpiece within the support area are consistent at each moment. Based on this hypothesis, the physical parameters at the center of the support area can be used to represent the entire support area, which is convenient for simplified analysis using the lumped parameter method.

[0079] Hypothesis 2: The support head does not undergo axial deformation and does not separate from the workpiece surface during movement.

[0080] Based on this hypothesis, the support head is split into two parts. The first part is the part of the support head that moves in contact with the workpiece, including the piston rod and the universal ball fixedly connected to it, etc. The mass of this part is denoted as ; The second part is the part of the support head installed on the robot flange, and its movement situation is consistent with that of the robot end. The mass of this part is denoted as . Splitting the support head in this way can be closer to the actual working conditions. At the same time, after bundling the masses, the contact forces between subsystems can be regarded as internal forces of the system and not considered, which simplifies the analysis steps.

[0081] In summary, a "mass-spring-damper" vibration model of the follow-up support area as shown in Figure 2 is established, which is a milling robot follow-up hybrid support structure. This structure is a three-stage hybrid type of "parallel-series-parallel", including a support point equivalent unit, a robot equivalent unit, and a workpiece supported area equivalent unit;

[0082] The support point equivalent units are first connected in parallel, and then the parallel-connected support point equivalent units are connected in series with the robot equivalent unit. The series-connected support point equivalent units, robot equivalent units, and the workpiece supported area equivalent system are then connected in parallel;

[0083] The support point equivalent unit includes support nodes. The support nodes are connected by a stud between a needle-shaped cylinder and a universal ball. The needle-shaped cylinder at one end of the support node is arranged on the flange surface of the robot equivalent unit, forming a mass bundling with the robot. The universal ball at the other end contacts the workpiece surface, forming a mass bundling with the workpiece. The two mass bundlings can regard the contact force between subsystems as internal forces of the system and not consider them, simplifying the analysis steps of subsequent modeling;

[0084] There is gas inside the needle-shaped cylinder of the support node.

[0085] Take , and construct the vibration equation as:

[0086] ;

[0087] Among them, ;

[0088] Among them, is time, is the mass matrix of the vibration equation, is the stiffness matrix of the vibration equation, is the damping matrix of the vibration equation, is the force vector of the vibration equation, is the displacement vector of the vibration equation, is the velocity vector of the vibration equation, is the acceleration vector of the vibration equation, is the axial milling force, is the total support force generated by air pressure, and are the displacements of the workpiece supported area and the robot flange along the normal direction of the workpiece respectively, are the equivalent mass, equivalent stiffness and equivalent damping of the workpiece supported area, are the equivalent mass, equivalent stiffness and equivalent damping of the robot, is the number of support points of the multi-point support head, are the equivalent stiffness and equivalent damping of a single support point.

[0089] Adopt the Rayleigh model to construct the system damping matrix, and express the damping matrix as a linear combination of the system mass matrix and stiffness matrix: ;

[0090] In the formula, is the equivalent proportional coefficient of the mass matrix, is the equivalent proportional coefficient of the stiffness matrix.

[0091] Then the vibration equation corresponding to the vibration modeling is:

[0092] ;

[0093] Compared with traditional profile support and multi-point array support, the follow-up support is a dynamic support strategy. During the support process, physical quantities such as the position of the workpiece support area, the pose of the robot end, and the air pressure of the support head are all time-varying, which leads to the stiffness value in vibration modeling is also time-varying. In order to more accurately predict and analyze the vibration characteristics of the follow-up support, it is necessary to focus on constructing a physical model of the time-varying hybrid stiffness field.

[0094] The stiffness of a thin-walled part refers to the ability to resist deformation at each position after boundary clamping. Since only the local stiffness of a specific sub-region of the workpiece needs to be obtained, the equivalent stiffness method of concentrated load in the plate and shell theory is used to obtain it. When examining the equivalent stiffness at the point, a concentrated load acting at the point can be applied to the right side of the workpiece deformation equation:

[0095] ;

[0096] In the formula, is the bending stiffness of the workpiece, is the normal deformation of the workpiece at the point, and is the Dirac function used to represent the position of the point load.

[0097] Solving the deformation from the above formula, the equivalent stiffness at the point can be calculated through the following formula, ;

[0098] However, solving the deformation at the specified position is time-consuming. Therefore, it is necessary to construct an explicit analytical formula of for convenient real-time substitution. In the present invention, the Galerkin method is used to obtain an approximate solution for typical cases.

[0099] Take the length of the rectangular thin plate as , the width as . Define the origin of the workpiece coordinate system at the center of the thin plate. The boundary conditions when the four sides are fixed can be written as:

[0100] ;

[0101] Use trigonometric series to construct a basis function that satisfies the boundary conditions:

[0102] ;

[0103] Then the approximate solution of can be written as a linear combination of the basis functions:

[0104] ;

[0105] wherein, is the coefficient to be solved.

[0106] Define the residual :

[0107] ;

[0108] Take the weight function to be consistent with the basis function and at the same time set in the above formula. At this time, there is only one coefficient to be solved. By making the inner product of the residual and the weight function equal to 0, the undetermined coefficient is obtained, that is:

[0109] ;

[0110] Expanding the above formula gives:

[0111] ;

[0112] Arrange it into two equal parts on the left and right:

[0113] ;

[0114] ;

[0115] For the right side, separate variables and combine the screening property of the Dirac function to get:

[0116] ;

[0117] For the left side, calculate the bi-Laplacian operator and combine the symmetry to get:

[0118] ;

[0119] Combining the left and right sides of the equation gives:

[0120] ;

[0121] Then the approximate equivalent stiffness at the required point is:

[0122] ;

[0123] Substitute it into the motion equation at the support center to obtain the time-varying approximate analytical stiffness value of the workpiece support domain :

[0124] ;

[0125] Among them, The abscissa of the point in the workpiece supported area, The ordinate of the point in the workpiece supported area, is the bending stiffness of the workpiece, is half of the workpiece length, is half of the workpiece width.

[0126] The equivalent stiffness of a single support point mainly comes from the "gas spring" inside the cylinder, that is, when the piston rod of the pointer-type cylinder is pushed out a preset distance later, there is a section of gas in the rodless cavity of the cylinder. When the workpiece unit vibrates, this section of gas will be stretched or compressed, forming an additional restoring force similar to a spring. This equivalent stiffness can be derived from the state equation of the gas in the adiabatic process:

[0127] ;

[0128] In the formula, is the air pressure in the rodless cavity of the support point cylinder, is the volume of the rodless cavity, is the adiabatic coefficient of the ideal gas, represents a constant constant.

[0129] Differentiating both sides of the above with respect to yields: ;

[0130] After rearrangement, we get: ;

[0131] Note that: ; ;

[0132] In the formula, is the cross-sectional area of the rodless cavity of the cylinder, is the gas pressure, .

[0133] Substituting Equation and Equation into Equation yields: ;

[0134] Note that the air pressure is a function of time, and after rearrangement according to the definition of stiffness, we get: .

[0135] During the follow-up support process, the end of the robot is always perpendicular to the workpiece surface. This invention focuses on equivalenting the stiffness of each joint of the robot to its end normal direction. The research object is a serial robot with degrees of freedom, and the generalized coordinates are , where is the rotation angle of the th joint of the robot.

[0136] First, establish the mapping relationship between the external force received at the end of the robot and the joint torque through the Jacobian matrix. All quantities are represented in the world coordinate system: ;

[0137] In the formula, is the vector composed of each joint torque, is the external force acting on the end of the robot, is the -dimensional linear velocity Jacobian matrix of the robot.

[0138] Let be the displacement information at the end of the robot. Taking the total derivative of both sides of Equation with respect to yields: ;

[0139] Combining the definitions of stiffness and the linear velocity Jacobian matrix, the above equation can be written as: ;

[0140] In the formula, is the joint stiffness matrix, the complementary stiffness matrix, which indicates that the Jacobian matrix of the robot changes due to external loads or its attitude. Under the selection of appropriate robot postures, compared with can be ignored. is the translational stiffness matrix equivalent to the end of the robot.

[0141] When and the time-varying information of the joint are known, the stiffness matrix can be calculated through the following formula:

[0142] ;

[0143] Through the rotation matrix between the base coordinate system and the end coordinate system of the robot, the stiffness matrix described in the end coordinate system is obtained:

[0144] ;

[0145] Yes is a matrix, and the elements on its diagonal represent the stiffness along the end coordinate system respectively, , and the stiffness in the three axial directions of . The non-diagonal elements represent the coupling stiffness between different directions. During the machining process of the follow-up support, the normal equivalent stiffness of the robot is the largest among the diagonal elements, denoted as

[0146] .

[0147] Embodiment 2

[0148] A control method for vibration modeling of a milling machining robot's follow-up hybrid support structure includes two steps: decoupling of the control equation and design of constraint following control.

[0149] During the machining process of the follow-up support, the trajectory of the support center and the joint states of the robot are both pre-planned and designed in advance. That is, the terms that can be actively controlled in the vibration equation are :

[0150] ;

[0151] The controllable terms appear on both the left and right sides of the equation simultaneously. That is, the change of will simultaneously affect the force and the vibration modeling parameters, which increases the complexity of the control design. For this reason, consider extracting and organizing all the terms related to

[0152] ;

[0153] In the formula:

[0154] ;

[0155] The above formula realizes the decoupling of the control terms and the follow-up hybrid support structure of the milling machining robot. Note that there is only one quantity for the control input, but there are two state variables to be controlled in the follow-up hybrid support structure of the milling machining robot and . At this time, the problem is transformed into the control design problem of an underactuated system, is the input matrix (vector) of the system. The underactuated system cannot meet all the response requirements of the system through a single control. Considering that the primary task of the follow-up support strategy is to suppress the vibration of the workpiece, the control task is to design so that meets the requirements.

[0156] Write the control objective in the form of servo constraints:

[0157] ;

[0158] In the formula, , is the objective actively designed by the controller to control the vibration amplitude to reduce the vibration energy according to specific cases, or to adjust the vibration frequency to avoid resonance.

[0159] Arrange it into matrix form: ;

[0160] In the formula: ;

[0161] Taking the time derivative on both sides can be rewritten as a second-order form of constraint: ;

[0162] In the formula: .

[0163] Assumption 3: For all , there is always .

[0164] This assumption is mainly to ensure that the Moore-Penrose generalized inverse of (denoted by the superscript "+" hereafter) exists and is unique. If the assumption is not satisfied, the control input of the system will have no effect on the state variables, and the present invention will avoid such a singular state through advance planning.

[0165] Assumption 4: For the given constant matrix and at the same time , there exists a constant , for all , ;

[0166] This assumption is used to ensure that the minimum eigenvalue of is always at a finite distance from 0, rather than approaching 0 infinitely.

[0167] Define the constraint following error: ;

[0168] Based on the constraint following theory, the present invention designs the following air pressure expression:

[0169] ;

[0170] Among them:

[0171] ;

[0172] ;

[0173] In the formula, is the control parameter, is the nominal control term, which is the generalization of the Udwadia-Kalaba principle in the underactuated system and has the minimum norm among all the control forces that satisfy the system servo motion requirements; is the feedback control term, which is used to compensate for the deviation between the actual state and the desired state of the system and is also designed for the underactuated system.

[0174] Theorem 1: The constraint following error satisfies: for any positive real number , there exists a finite positive real number , if , then for all , there is (Lyapunov stability).

[0175] Let: ;

[0176] For This is a legal Lyapunov candidate function, and the result of its derivative is as follows:

[0177] ;

[0178] The present invention divides into two parts. For part1, there is:

[0179] ;

[0180] For part2, there is:

[0181] ;

[0182] Combining them gives: .

[0183] Example 3

[0184] Time-varying stiffness field analysis:

[0185] Now, the characteristics of the three main stiffnesses of the follow-up support stiffness field are analyzed through simulation. Take , , and plot As shown in Figure 3 , it can be seen that for the thin flat plate workpiece clamped and fixed on four sides, its stiffness has a significant annular attenuation trend from the boundary to the center, and the order of magnitude of the weakest stiffness is in the range of . Take , , and plot As shown in Figure 4As shown, for a single support node, its equivalent stiffness is inversely proportional to the cylinder extension distance and directly proportional to the input air pressure . The overall stiffness magnitude is generally within the magnitude range. After paralleling multiple support points, the total stiffness provided can reach the magnitude range, which is consistent with the stiffness level at the weak part of the workpiece.

[0186] Taking a 6-degree-of-freedom serial manipulator for simulation analysis, its joint stiffness matrix is . Plot as shown in Figure 5 . In the robot's workspace, most of its stiffness values are evenly distributed within the range. The closer to the boundary of the workspace, the greater this stiffness value, but the flexibility of the robot will be significantly reduced. Therefore, when actually machining, it is necessary to comprehensively consider the stiffness and flexibility indicators to determine the relative position between the robot and the workpiece.

[0187] In various analyses, the stiffness of the manipulator at the boundary of the workspace is often the weakest, but the results of the present invention do not conflict with the above classical conclusions. This is because the stiffness of the present invention has a clear directionality rather than considering the global stiffness. When the robot moves to the boundary of the workspace, its various linkages will be close to a collinear state. At this time, the manipulator can be analogized to a "cantilever beam" system. For a cantilever beam, when a non-axial force is applied to its end, a large bending deformation will occur. Therefore, when evaluating the comprehensive stiffness index, it will show a low bending resistance performance. However, the present invention focuses on the anti-deformation ability along the axial direction of the end. At this time, it is similar to stretching or compressing along the axial direction of the entire beam, so the stiffness performance shown is stronger.

[0188] In summary, the three main coupling stiffnesses of the follow-up support are all complex and time-varying, and generally of the same order of magnitude. To maintain a stable and good support effect, it is necessary to carry out control design in combination with the dynamic parameter changes during processing.

[0189] The present invention will start from the overall vibration dynamics modeling of the system and analyze the effectiveness of the constraint following strategy for vibration control. Take , the mass parameter is , the number of support points , and the fixed extension distance of the support points is . Set the external excitation force , and at the same time, the movement path in the workpiece coordinate system is to simulate a machining case where both the magnitude and position of the acting force are time-varying.

[0190] To more intuitively reflect the advantages of the constraint following control (CFC) proposed by the present invention, the present invention selects feedback linearization control (FLC) and PD as comparative control strategies. The expression of FLC is divided into: ;

[0191] Meanwhile, there is: ;

[0192] In the formula, is the differential gain matrix, is the proportional gain matrix.

[0193] The expression of PD is: ;

[0194] In the formula, is the differential gain coefficient, is the proportional gain coefficient.

[0195] Take the tracking target as , for analysis. As Figure 6 shown, the vibration displacement can approach the desired function within 1 s under different algorithms. The displacement curve of the CFC method is smoother than the other two curves and can reach a control accuracy close to the order of magnitude. In contrast, the PD method has more significant oscillations near the desired trajectory, and the achieved stable control accuracy is of the order of magnitude. Define the total position error as , as Figure 7 shown, both CFC and FLC based on modeling are lower than the non-modeling method PD. Among them, the total error of the CFC method is the smallest, which is about 43% lower than the PD method.

[0196] Plot the control term under the three methods. As Figure 8 shown, at the beginning of the control, both PD and FLC produce large oscillations. The CFC method has relatively stable control. At the same time, the magnitude of under the three methods in the stable stage is all near . Define the total control cost as , as Figure 9 shown, both PD and CFC show the advantage of low cost. Among them, the control cost of CFC is about 54% lower than that of FLC.

[0197] In summary, the CFC algorithm proposed by the present invention can achieve stable tracking of the desired vibration effect with the combined advantages of high precision and low cost.

[0198] Finally, it should be noted that the above embodiments are only used to illustrate the technical method of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify the technical method of the present invention or make equivalent substitutions, and these modifications or equivalent substitutions cannot make the modified technical method deviate from the spirit and scope of the technical method of the present invention.

Claims

1. A vibration modeling method for a milling robot follow-up hybrid support structure, the vibration equation corresponding to the vibration modeling method is: ； ； ； ； in, For time, is the mass matrix of the vibration equation, is the stiffness matrix of the vibration equation, is the force vector of the vibration equation, is the displacement vector of the vibration equation, is the velocity vector of the vibration equation, is the acceleration vector of the vibration equation, is the axial milling force, is the total support force generated by air pressure, and are the displacements of the supported area of ​​the workpiece and the robot flange along the normal direction of the workpiece, is the equivalent mass and equivalent stiffness of the supported area of ​​the workpiece, is the equivalent mass and equivalent stiffness of the robot, is the number of support points of the multi-point support head, is the equivalent stiffness of a single support point, To support the mass of the binding part between the node and the workpiece, To support the mass of the node and the robot bundle, The damping matrix for the equivalent vibration equation, is the equivalent proportional coefficient of the mass matrix, is the equivalent proportional coefficient of the stiffness matrix; The milling robot follow-up hybrid support structure is a "parallel-series-parallel" three-level hybrid configuration, including a support point equivalent unit, a robot equivalent unit and a workpiece supported area equivalent unit; The support point equivalent units are first connected in parallel, and the parallel-connected support point equivalent units are then connected in series with the robot equivalent units. The series-connected support point equivalent units and the robot equivalent units are then connected in parallel with the workpiece supported area equivalent system; The support point equivalent unit includes a support node, and the support node is connected by a needle cylinder and a universal ball through a stud. The needle cylinder at one end of the support node is arranged on the flange surface of the robot equivalent unit to form a mass binding with the robot, and the universal ball at the other end contacts the surface of the workpiece to form a mass binding with the workpiece; Gas is arranged inside the needle-shaped cylinder of the supporting node.

2. The vibration modeling method of a milling robot follow-up hybrid support structure according to claim 1 is characterized in that: The equivalent stiffness expression of the supported area of ​​the workpiece is: ； in, The horizontal coordinate of the point in the supported area of ​​the workpiece, The ordinate of the point in the supported area of ​​the workpiece, is the bending stiffness of the workpiece, is half the length of the workpiece, Half the width of the workpiece.

3. The vibration modeling method of a milling robot follow-up hybrid support structure according to claim 1, characterized in that: The degree of freedom of the serial robot, the generalized coordinates are ,in For the robot The equivalent stiffness matrix of the joint angle and robot normal is: ； in, is the rotation matrix between the robot base coordinate system and the end coordinate system, for dimensional robot linear velocity Jacobian matrix, is the joint stiffness matrix, Used to obtain the maximum value of the diagonal elements of the result of the operation within the brackets.

4. The vibration modeling method of a milling robot follower hybrid support structure according to claim 1, characterized in that: The equivalent stiffness expression of a single support point is: ； in, is the air pressure in the rodless chamber of the supporting cylinder. is the ideal gas adiabatic coefficient, is the cross-sectional area of ​​the rodless chamber of the cylinder, It is the preset distance that the piston rod of the needle type cylinder extends.

5. A control method for vibration modeling of a milling robot follow-up hybrid support structure, characterized in that: It includes two steps: decoupling of control equations and designing constraint following control; The specific content of the control equation decoupling is: Active Control The vibration equation corresponding to the vibration modeling is decoupled to obtain the equivalent control equation: ； Where: ； is the mass matrix of the vibration equation, is the acceleration vector of the vibration equation, are the damping matrix, stiffness matrix and force vector in the equivalent control equation, respectively. is the velocity vector of the vibration equation, is the displacement vector of the vibration equation, is the air pressure in the rodless chamber of the supporting cylinder. and are two state variables to be controlled, then define is the input matrix or input vector of the milling robot follow-up hybrid support structure. is the equivalent proportional coefficient of the mass matrix, is the equivalent proportional coefficient of the stiffness matrix, is the equivalent mass and equivalent stiffness of the supported area of ​​the workpiece, To support the mass of the binding part between the node and the workpiece, is the equivalent mass and equivalent stiffness of the robot, To support the mass of the node and the robot bundle, is the axial milling force, is the number of support points of the multi-point support head, is the cross-sectional area of ​​the rodless chamber of the cylinder, is the ideal gas adiabatic coefficient, The preset distance of the needle cylinder piston rod extension. and are the displacements of the supported area of ​​the workpiece and the robot flange along the normal direction of the workpiece, and are the velocities of the supported area of ​​the workpiece and the robot flange along the normal direction of the workpiece, respectively; Control tasks are designed ,make Meet the needs.

6. The control method for vibration modeling of a milling robot follower hybrid support structure according to claim 5, characterized in that: include: Will control the target Written in the form of a servo constraint: ； In the formula, , Actively designed goals for the controller to control the vibration amplitude to reduce the vibration energy or adjust the vibration frequency to avoid resonance according to the specific case; Will control the target Arranged into matrix form: Where: ； Will control the target The time derivatives of both sides of the matrix form are rewritten as second-order constraints: ； Where: ； For all , both have ; For a given constant matrix at the same time , there is a constant , for all , ; Define the constrained following error: ; Air pressure expression: ; in: ; ； In the formula, To control the parameters, is the nominal control term, It is the generalization of the Udwadia-Kalaba principle to underactuated systems, and it has the smallest norm among all control forces that meet the servo motion requirements of the system; It is a feedback control term used to compensate for the deviation between the actual state of the system and the desired state; Constrained Following Error Satisfies: For any positive real number , there exists a finite positive real number ,if , then for all ,have , .

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