Milling machining robot follow-up series-parallel supporting structure, vibration modeling and control method of milling machining robot follow-up series-parallel supporting structure

Abstract

The invention discloses a milling robot follow-up series-parallel supporting structure, a vibration modeling method and a control method thereof, and relates to the field of milling. The milling robot follow-up series-parallel supporting structure is of a three-stage series-parallel configuration; comprising a supporting point equivalent unit, a robot equivalent unit and a workpiece supported area equivalent unit. The supporting point equivalent units are firstly connected in parallel, the supporting point equivalent units connected in parallel are connected in series with the robot equivalent unit, and the supporting point equivalent units connected in series and the robot equivalent unit are connected in parallel with the workpiece supported area equivalent system; the vibration modeling comprises a vibration equation corresponding to the vibration modeling, and the control method comprises two steps of control equation decoupling and design constraint following control. According to the milling robot follow-up series-parallel supporting structure adopting the robot follow-up supporting strategy and the constraint follow-up control method of the milling robot follow-up series-parallel supporting structure, the servo requirement of vibration control can be met 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 of a milling processing robot, vibration modeling and control method thereof. Background Art

[0002] Thin-walled parts are widely used in fields such as aviation and aerospace, but their weak rigidity makes them prone to deformation and vibration during processing, further affecting the processing accuracy and surface quality. The application of appropriate support strategies can effectively solve this problem. The three mainstream support strategies are the formwork support strategy, the multi-point discrete support strategy, and the follower support strategy. Among them, the formwork support strategy can provide a stable support effect, but it is expensive and has poor compatibility. In this regard, the multi-point discrete support strategy can adapt to components of different geometric shapes by adjusting the height and angle of each support node, but there are gaps between each node, and it is difficult to ensure local stiffness and processing accuracy in the gaps. The follower support strategy can provide low-cost, high-flexibility support effects for the processing of thin-walled workpieces, effectively circumventing the above problems. This strategy drives the support head to follow the processing tool for coordinated movement, which can save costs while ensuring that sufficient local stiffness is always maintained at the source of the excitation.

[0003] Patent CN110332277B proposes a vibration control device for curved thin-walled parts based on permanent magnet drive. One drive source can achieve support in multiple positions, but there are large gaps between support nodes. Compared with the robot's follow-up support, local stiffness is difficult to guarantee.

[0004] Patent CN114248134B proposes a method for adjusting vibration suppression using a follower support suitable for machining thin-walled parts of varying thicknesses. The vibration suppression mechanism comprises an active and passive device, with the active-side telescopic mechanism providing vibration suppression tailored to varying thicknesses and milling depths. While this method accounts for variations in workpiece thickness, it does not account for changes in stiffness caused by changes in the actuator's position.

[0005] To describe and predict the deformation and vibration of a workpiece under a servo support strategy, a dynamic model of the servo support strategy is necessary. This also provides a theoretical basis for the subsequent optimization design of the support effect. Existing research on servo support strategies has not yet fully considered the relationship between the coupled vibration and stiffness of the three heterogeneous systems: the workpiece, the support head, and the robot.

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

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

[0008] Among them, a milling robot follow-up hybrid support structure, 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, which is connected by a needle cylinder and a universal ball through a stud. The needle cylinder at one end of the support node is set 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 provided inside the needle-shaped cylinder of the support node.

[0009] A vibration modeling of a milling robot follow-up hybrid support structure, the vibration equation corresponding to the vibration modeling 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 supporting force generated by air pressure, and are the displacements of the workpiece support area and the robot flange along the workpiece normal direction, are the equivalent mass and equivalent stiffness of the supported area of ​​the workpiece, are 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 binding part between the node and the robot, 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.

[0010] Preferably, 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 vertical coordinate 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.

[0011] Preferably, The degree of freedom serial robot, the generalized coordinates are ,in For the robot The equivalent stiffness matrix of the joint angle and the 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.

[0012] Preferably, 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.

[0013] A vibration modeling control method for a milling robot follower hybrid support structure includes two steps: control equation decoupling and design constraint following control.

[0014] Preferably, 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: ; 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. are the damping matrix, stiffness matrix and force vector in the equivalent control equation respectively; Control task is designed ,make Meet the needs.

[0015] Preferably, it includes: Control the target Written in the form of servo constraints: ; Where, , Actively designed goals for the controller to control the vibration amplitude to reduce vibration energy or adjust the vibration frequency to avoid resonance according to specific cases; Control the target Arranged into matrix form: ; Where: ; Control the target The time derivatives on 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: ; ; Where, For the control parameters, is the nominal control item, It is a 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 and the desired state of the system; Constrained following error Satisfies: For any positive real number , there exists a finite positive real number ,if , then for all ,have , .

[0016] In summary, the present invention provides a milling robot follower hybrid support structure, vibration modeling, and control method thereof. Compared with conventional technologies, the present invention proposes a "parallel-series-parallel" three-stage hybrid follower support vibration configuration. This configuration fully considers the time-varying stiffness field characteristics of the three heterogeneous systems of the workpiece, support head, and robot, and provides an analytical model of each stiffness. A control method for vibration modeling of a milling robot follower hybrid support structure is proposed. This method is based on an active vibration control strategy for the follower support system based on the constraint following theory. This strategy decouples the controllable items of the system to obtain an equivalent underdriven control model. At the same time, the Lyapunov stability proof and comparative simulation verify 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.

[0017] The technical method of the present invention is further described in detail below through the accompanying drawings and examples. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 Diagrams of three mainstream thin-walled parts support strategies; Figure 2 This is a schematic diagram of the three-stage hybrid configuration of the follower support domain of the present invention; Figure 3 The distribution of equivalent stiffness of the workpiece at different positions of the workpiece; Figure 4 The relationship between the equivalent stiffness of the support point, the extension distance of the cylinder and the input air pressure; Figure 5 Distribution of normal equivalent stiffness of support points in the workspace; Figure 6 Workpiece vibration displacement under different control methods; Figure 7 Comparison of total error of workpiece vibration displacement under different control methods; Figure 8 Air pressure under different control methods; Figure 9 Comparison of total air pressure cost under different control methods.

[0019] Reference numerals 1. Support node; 11. Needle cylinder; 12. Universal ball; 2. Workpiece; 3. Gas; 4. Flange. DETAILED DESCRIPTION

[0020] The technical method of the present invention is further described below through the accompanying drawings and embodiments. It should be noted that unless otherwise specifically stated, the relative arrangement of components and steps, numerical expressions and values ​​described in these embodiments do not limit the scope of this application.

[0021] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the present disclosure, its application, or uses.

[0022] Technologies, systems, and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but where appropriate, they should be considered part of the specification.

[0023] In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not limiting. Therefore, other examples of the exemplary embodiments may have different values.

[0024] Unless otherwise defined, technical or scientific terms used in the present invention shall have the same meaning as commonly understood by one of ordinary skill in the art to which the present invention belongs.

[0025] like Figure 1 As shown in Figure 1, the three mainstream support strategies are: the model support strategy, the multi-point discrete support strategy, and the follower support strategy. These three support strategies are expensive, have poor compatibility, are difficult to ensure local stiffness and machining accuracy in the gaps, and fail to fully consider the relationship between coupled vibration and stiffness of the three heterogeneous systems: the workpiece, the support head, and the robot.

[0026] The present invention provides a follow-up hybrid support structure, vibration modeling and control method for a milling robot. First, a second-order vibration model of the follow-up support strategy is constructed, which simplifies the modeling through mass bundling while capturing the main stiffness characteristics of the system. Secondly, the time-varying stiffness analytical expressions of the three main stiffnesses of the local area of ​​the workpiece, the gas spring of the support point and the normal of the robot end are established through the elastic deformation equation, the adiabatic gas state equation and the robot force-Jacobi mapping, which facilitates the real-time input calculation in the control process. Finally, the controllable air pressure term is decoupled from both sides of the vibration equation to obtain an equivalent under-actuated system, and a constrained following control algorithm is designed. Comparative simulation shows that the proposed control method can achieve the servo requirements of vibration control with high precision and low cost.

[0027] Example 1 To simplify the modeling and capture the system's primary vibration characteristics, this paper focuses on vibration modeling within the support region that follows the excitation source, rather than focusing on vibration at all locations across the entire workpiece surface. Furthermore, because the thickness of thin-walled parts is much smaller than the other two dimensions, the workpiece's normal stiffness is weakest, resulting in more severe deformation and vibration. Therefore, this paper's modeling focuses on vibration along the workpiece normal direction.

[0028] Assumption 1: The vibration of the workpiece within the support area is consistent at every moment.

[0029] Because the support domain is very small relative to the entire thin-walled component, it can be assumed that the physical parameters of the workpiece within the support domain are consistent with the vibration conditions at each moment. Based on this assumption, the physical parameters at the center of the support domain can be used to represent the entire support domain, facilitating simplified analysis using the lumped parameter method.

[0030] Assumption 2: The support head does not undergo axial deformation and does not separate from the workpiece surface during movement.

[0031] Based on this assumption, the support head is divided 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 connected to it. The mass of this part is recorded as The second part is the part where the support head is installed on the robot flange. It is consistent with the movement of the robot end. The mass of this part is recorded as Splitting the support head in this way can better reflect the actual working conditions. At the same time, after bundling the mass, the contact forces between subsystems can be regarded as internal forces of the system and ignored, thus simplifying the analysis steps.

[0032] In summary, the establishment of Figure 2 The "mass-spring-damper" vibration model of the follower support domain shown is a milling robot follower hybrid support structure. This structure is a three-level "parallel-series-parallel" hybrid structure, including support point equivalent units, robot equivalent units, and workpiece supported area equivalent units. 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, which is connected by a needle cylinder and a universal ball via a stud. The needle cylinder at one end of the support node is set on the flange surface of the robot equivalent unit to form a mass bundle with the robot, and the universal ball at the other end contacts the workpiece surface to form a mass bundle with the workpiece. The mass bundle at the two locations can treat the contact force between the subsystems as an internal force of the system and ignore it, simplifying the analysis steps of subsequent modeling; Gas is provided inside the needle-shaped cylinder of the support node.

[0033] Pick , construct the vibration equation as: ; in, ; in, For 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 supporting force generated by air pressure, and are the displacements of the workpiece support area and the robot flange along the workpiece normal direction, are the equivalent mass, equivalent stiffness and equivalent damping of the supported area of ​​the workpiece, are the equivalent mass, equivalent stiffness and equivalent damping of the robot, is the number of support points of the multi-point support head, is the equivalent stiffness and equivalent damping of a single support point.

[0034] The Rayleigh model is used to construct the system damping matrix, which is expressed as a linear combination of the system mass matrix and stiffness matrix: ; Where, is the equivalent proportional coefficient of the mass matrix, is the equivalent proportional coefficient of the stiffness matrix.

[0035] The vibration equation corresponding to the vibration modeling is: ; Compared with traditional support and multi-point array support, follower support is a dynamic support strategy. During the support process, the position of the workpiece support area, the end posture of the robot, and the air pressure of the support head are all time-varying, which leads to the stiffness value in vibration modeling. In order to more accurately predict and analyze the vibration characteristics of the follower support, it is necessary to focus on constructing a physical model of the time-varying hybrid stiffness field.

[0036] The stiffness of thin-walled parts refers to the ability of each position to resist deformation after boundary clamping. Since only the local stiffness of a specific sub-region of the workpiece is required, the concentrated load equivalent stiffness method in plate and shell theory is used to obtain it. The equivalent stiffness at point , can be applied to the right side of the workpiece deformation equation Concentrated load at : ; Where, is the bending stiffness of the workpiece, For workpiece The normal deformation at is the Dirac function used to represent the position of point load.

[0037] The deformation can be obtained from the above formula , you can calculate the point by the following formula The equivalent stiffness at ; However, it is time-consuming to solve the deformation at a specified position, so it is necessary to build The explicit analytical expression of is used to facilitate real-time substitution. The present invention adopts the Galerkin method to obtain approximate solutions for typical cases.

[0038] Take the length of the rectangular plate as , width is , the origin of the workpiece coordinate system is defined to be located at the center of the thin plate, and the boundary conditions when the four sides are fixed can be written as: ; Use trigonometric series to construct basis functions that satisfy the boundary conditions: ; but Approximate solution of It can be written as a linear combination of basis functions: ; Where, are the coefficients to be solved.

[0039] Defining residuals : ; Weight function With basis functions Consistent, let , there is only one coefficient To be solved, by making the residual and weight function The inner product is 0 to obtain the unknown coefficients, that is: ; Expanding the above formula, we can get: ; Arrange it into two equal parts: ; ; For the right side, separate the variables and combine the Dirac The screening properties of the function are: ; For the left side, computing the double Laplacian operator and combining the symmetry gives: ; The combined left and right sides can be obtained: ; The required points The approximate equivalent stiffness at is: ; Substitute the equation of motion of the support center into , the time-varying approximate analytical stiffness value of the workpiece support domain can be obtained : ; in, The horizontal coordinate of the point in the supported area of ​​the workpiece, The vertical coordinate 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.

[0040] Equivalent stiffness of a single support point It mainly comes from the "gas spring" inside the cylinder, that is, the piston rod of the pointer-type cylinder pushes out the preset distance After the workpiece is energized, there is a section of gas in the rodless chamber of the cylinder. When the workpiece unit vibrates, this section of gas is 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: ; Where, is the air pressure in the rodless chamber of the supporting cylinder. is the volume of the rodless cavity, is the ideal gas adiabatic coefficient, Represents a constant.

[0041] Align both sides at the same time Taking the derivative we get: ; After finishing, we can get: ; Note: ; ; Where, is the cross-sectional area of ​​the rodless chamber of the cylinder, is the gas pressure, .

[0042] General Japanese style Substitution We can get: ; Noting that the air pressure is a function of time, and rearranging it according to the definition of stiffness, we can obtain: .

[0043] During the follow-up support process, the end of the robot is always perpendicular to the workpiece surface. This paper focuses on making the stiffness of each joint of the robot equivalent to its end normal. The degree of freedom serial robot, the generalized coordinates are ,in For the robot The rotation angle of the joint.

[0044] First, the Jacobian matrix is ​​used to establish the mapping relationship between the external force on the robot end and the joint torque. All quantities are uniformly expressed in the world coordinate system: ; Where, is the vector of joint torques, is the external force acting on the end of the robot, for dimensional Jacobian matrix of the robot's linear velocity.

[0045] make is the displacement information of the robot end, Both sides of the equal sign Finding the total derivative yields: ; Combining the definitions of stiffness and linear velocity Jacobian matrices, the above formula can be written as: ; Where, is the joint stiffness matrix, The complementary stiffness matrix, which is expressed as the Jacobian matrix of the robot subjected to external loads or changes with its posture, is selected under the appropriate robot posture. Compared to It can be ignored. is the translation stiffness matrix equivalent to the end of the robot.

[0046] In the known and joint time-varying information The stiffness matrix can be calculated by the following formula : ; By the rotation matrix between the robot base coordinate system and the end coordinate system , get the coordinate system at the end The stiffness matrix described below : ; yes The matrix, whose diagonal elements represent the coordinates along the end of 、 and The stiffness in the three axis directions, and the non-diagonal elements represent the coupling stiffness between different directions. During the follower support processing, the robot normal equivalent stiffness is the largest among the diagonal elements, which is recorded as : .

[0047] Example 2 A vibration modeling control method for a milling robot follower hybrid support structure includes two steps: control equation decoupling and design constraint following control.

[0048] During the follow-up support processing, the trajectory of the support center and the robot's joint states All are planned and designed in advance, that is, the items that can be actively controlled in the vibration equation are : ; The controllable terms appear on both the left and right sides of the equation, i.e. The change of will affect the vibration modeling parameters at the same time, which increases the complexity of control design. The relevant terms are extracted and sorted to obtain the following equivalent system equations: ; Where: ; The above formula realizes the decoupling of the control term and the follow-up hybrid support structure of the milling robot. Note that the control input is only One quantity, but there are two state variables to be controlled in the milling robot follow-up hybrid support structure and , then the problem is transformed into the control design problem of 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 ,make Meet the needs.

[0049] The control objective is written in the form of servo constraints: ; Where, , Actively designed goals for the controller to control the vibration amplitude to reduce vibration energy or adjust the vibration frequency to avoid resonance according to specific cases.

[0050] Arranged into matrix form: ; Where: ; The time derivatives on both sides can be rewritten as second-order constraints: ; Where: .

[0051] Assumption 3: For all , both have .

[0052] This assumption is mainly to ensure The Moore-Penrose generalized inverse (denoted by a superscript "+") of exists and is unique. If this assumption is not met, the system's control input will have no effect on the state variables. The present invention avoids this singular state through advance planning.

[0053] Assumption 4: For a given constant matrix at the same time , there is a constant , for all , ; This assumption is used to ensure The minimum eigenvalue of is always finitely far away from 0, rather than infinitely close to 0.

[0054] Define the constrained following error: ; Based on the constraint following theory, the present invention designs the following air pressure expression: ; in: ; ; Where, For the control parameters, is the nominal control term, which is a generalization of the Udwadia-Kalaba principle to underactuated systems and has the smallest norm among all control forces that meet the servo motion requirements of the system; It is a feedback control item used to compensate for the deviation between the actual state and the desired state of the system. It is also designed for under-actuated systems.

[0055] Theorem 1: Constrained Following Error Satisfies: For any positive real number , there exists a finite positive real number ,if , then for all ,have (Lyapunov stability).

[0056] make: ; for This is a valid Lyapunov candidate function, and the results of its derivation are as follows: ; The present invention will It is divided into two parts, for part 1 there are: ; For part2, we have: ; Jointly: .

[0057] Example 3 Time-varying stiffness field analysis: The characteristics of the three main stiffnesses of the follow-up support stiffness field are analyzed through simulation. , ,draw like Figure 3 As shown in the figure, it can be seen that for a thin flat workpiece clamped 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 Within the range. , ,draw like Figure 4 As shown in the figure, for a single support node, its equivalent stiffness is proportional to the cylinder extension distance. Inversely proportional to the input pressure is proportional to the overall stiffness level. After connecting multiple support points in parallel, the total stiffness can reach This is consistent with the stiffness level of the weak points of the workpiece.

[0058] Take a 6-DOF serial manipulator for simulation analysis, and its joint stiffness matrix is ,draw like Figure 5 As shown in Figure 2, within the robot workspace, most of its stiffness values ​​are within The stiffness is evenly distributed within the range. The closer to the boundary of the workspace, the greater the stiffness value. However, the flexibility of the robot will be significantly reduced. Therefore, in actual processing, the relative position between the robot and the workpiece must be determined by comprehensively considering the stiffness and flexibility indicators.

[0059] In various analyses, the stiffness of a robotic arm at the boundary of its workspace is often the weakest, but the results of the present invention do not contradict these classic conclusions. This is because the stiffness of the present invention has a clear directionality rather than considering global stiffness. When the robot moves to the boundary of the workspace, its various links will approach a collinear state. At this time, the robotic arm can be compared to a "cantilever beam" system. For a cantilever beam, its end will produce large bending deformation when subjected to non-axial forces, so it will show low bending resistance when evaluating the comprehensive stiffness index. However, the present invention focuses on the deformation resistance along the axial direction of the end, which is similar to pulling or compressing along the axial direction of the entire beam, so the stiffness performance is stronger.

[0060] In summary, the three main coupling stiffnesses of the follower support are all complex and time-varying, and at the same time, they are of comparable magnitude. In order to maintain a stable and good support effect, it is necessary to carry out control design in combination with the changes in dynamic parameters during processing.

[0061] This paper will start from the overall vibration dynamics modeling of the system and analyze the effectiveness of the constraint following strategy for vibration control. , the quality parameter is , number of support points , the support point extends a fixed distance . Set external excitation force , and the moving path in the workpiece coordinate system is , which is used to simulate machining cases where both the magnitude and position of the applied force vary with time.

[0062] In order to more intuitively reflect the advantages of the constrained following control (CFC) proposed in the present invention, the present invention selects feedback linearization control (FLC) and PD as comparative control strategies, where the expression of FLC is divided into: ; Also available: ; Where, is the differential gain matrix, is the proportional gain matrix.

[0063] The expression of PD is: ; Where, is the differential gain coefficient, is the proportional gain coefficient.

[0064] Take the tracking target as , For analysis. Figure 6 As shown, the vibration displacement Under different algorithms, the expected function can be approximated within 1s. The displacement curve of the CFC method is smoother than the other two curves, and can reach close to In contrast, the PD method has a more significant oscillation near the desired trajectory, and the stable control accuracy achieved is The total position error is defined as ,like Figure 7 As shown in Figure 3, both the modeling-based CFC and FLC are lower than the modeling-free method PD, among which the total error of the CFC method is the smallest, which is reduced by about 43% compared with the PD method.

[0065] Drawing controls in three ways like Figure 8 As shown in the figure, during the initial control period, both PD and FLC produced large oscillations. The CFC method was relatively stable in control. At the same time, the three methods in the stable stage The magnitude of The total control cost is defined as ,like Figure 9 As shown in the figure, both PD and CFC have the advantage of low cost, among which the control cost of CFC is reduced by about 54% compared with FLC.

[0066] In summary, the CFC algorithm proposed in the present invention can achieve stable tracking of the desired vibration effect with the combined advantages of high precision and low cost.

[0067] 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 skilled in the art should understand that they can still modify or replace the technical method of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical method to deviate from the spirit and scope of the technical method of the present invention.

Claims

1. A milling robot follow-up hybrid support structure, characterized in that: 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. A vibration modeling of a milling robot follow-up hybrid support structure, characterized in that: The vibration equation corresponding to the vibration modeling 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.

3. The vibration modeling of the follow-up hybrid support structure of a milling robot according to claim 2 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.

4. The vibration modeling of the follow-up hybrid support structure of a milling robot according to claim 3 is 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 rotation 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.

5. The vibration modeling of the follow-up hybrid support structure of a milling robot according to claim 4 is 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.

6. 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.

7. The control method for vibration modeling of a milling robot follower hybrid support structure according to claim 6, characterized in that: 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: ； 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. are the damping matrix, stiffness matrix and force vector in the equivalent control equations respectively; Control tasks are designed ,make Meet the needs.

8. The control method for vibration modeling of a milling robot follower hybrid support structure according to claim 7, 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 constraint 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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