Mechanical arm milling stability prediction method

Abstract

The present application relates to a kind of mechanical arm milling stability prediction method, the method is first to carry out axial region division to cutting area;Then the static cutting region divided with dynamic cutting region is not equal distance scattered into microelement, the window function and time lag coefficient required for tool microelement and workpiece contact judgment standard are calculated;Finally, the dynamic cutting force matrix and milling process multi-time lag dynamics model stability solution obtained are used, complete mechanical arm milling stability prediction.Compared with the literature, the present application simultaneously considers the contact relationship of tool-workpiece caused by axial low-frequency vibration and radial low-frequency vibration to establish the stability prediction method, can solve the influence of severe low-frequency vibration on machining process during mechanical arm milling, improve the prediction accuracy of mechanical arm milling stability petal diagram under weak stiffness configuration.

Description

Technical Field

[0001] This invention belongs to the field of machining technology and relates to a method for predicting the milling stability of a robotic arm, particularly a method for predicting the milling stability of a robotic arm with a weak stiffness configuration. Background Technology

[0002] Traditional CNC machining centers have much higher machining stiffness than robotic arm milling, resulting in lower stability in robotic arm milling compared to CNC machining centers. However, the amplitude of low-frequency structural vibrations generated by the weak-stiffness configuration is on the same order of magnitude as the feed per tooth, often leading to tool-workpiece separation during actual robotic arm milling. Therefore, robotic arm milling involves both radial and axial tool-workpiece separation, and the effects of both must be considered when predicting the stability of robotic arm milling.

[0003] Reference 1, "Y. Mohammadi, K. Ahmadi, Effect of axial vibrations on regenerative chatter in robotic milling, Procedia CIRP 82(2019)503-508," discloses a cutting force model suitable for robotic arm milling. This model first considers incorporating the modulation effect of axial vibration on the depth of cut into the dynamic model, then uses a traditional dynamic model to compensate for the generated stiffness-like and nonlinear terms, and finally uses a stability leaflet diagram to complete the prediction of robotic arm milling. However, this method does not consider the impact of low-frequency vibrations caused by the weak stiffness of the robotic arm structure on stability prediction.

[0004] Reference 2, “SHXin, FYPeng, XWTang, R.Yan, ZPLi, JWWu, Research on the influence of robot structural mode on regenerative chatter in milling and analysis of stability boundary improvement domain, International Journal of Machine Tools and Manufacture 179(2022)103918,” discloses a dynamic model suitable for robot arm milling. This model incorporates displacement compensation caused by low-frequency vibration into the dynamic model and finally calculates the stability lobe diagram considering radial low-frequency vibration. This method only analyzes the radial tool-workpiece separation phenomenon and does not consider the influence of axial low-frequency vibration on stability prediction.

[0005] The typical characteristic of the aforementioned references is that they only consider the effects of regenerative chatter or radial forced vibration on robotic arm milling stability prediction, neglecting the tool-workpiece separation phenomenon caused by low-frequency axial vibration. However, during robotic arm milling, axial vibration generates severe vibration phenomena and significantly affects the accuracy of robotic arm milling stability prediction. Summary of the Invention

[0006] Technical problems to be solved

[0007] To overcome the shortcomings of existing methods that fail to consider low-frequency axial structural vibrations in predicting the stability of robotic arms during milling, and thus cannot properly model the milling process under the weak stiffness configuration of the robotic arm, this invention proposes a milling stability prediction method that is applicable to the milling process of robotic arms and considers both radial and axial low-frequency vibrations.

[0008] Technical solution

[0009] A method for predicting the stability of a robotic arm during milling, characterized by the following steps:

[0010] Step 1: Calibrate the tangential, radial, and axial cutting force coefficients through milling experiments;

[0011] Step 2: Modal impact tests were conducted on the tool and robotic arm structure using the standard force hammer impact method to obtain the natural frequency matrix ω of the tool and robotic arm structure. n,c ω n,str Damping ratio matrix ζ c ζ str and modal mass matrix m c m str Where ω n,c Let ω be the natural frequency matrix of the cutting tool. n,str Let ζ be the natural frequency matrix of the robotic arm structure. c Let ζ be the tool damping ratio matrix. str Let m be the damping ratio matrix of the robotic arm structure. c Let m be the tool quality matrix. str The damping ratio matrix of the robotic arm structure;

[0012] Step 3: Divide the workpiece cutting area into a static cutting area and a dynamic cutting area according to the following formula, and then distribute them at unequal distances:

[0013]

[0014] Where w(t) is the actual axial cutting depth, w s w represents the axial cutting depth in the static cutting region. d (t) represents the axial cutting depth in the dynamic cutting region, wset For the set axial cutting depth, A ap A represents the maximum absolute amplitude of the axial low-frequency vibration. lv This is the difference between the maximum and minimum axial cutting depths caused by low-frequency vibration.

[0015] Step 4: Analyze the modeling of the static and dynamic cutting regions, and calculate the window function g of the axial micro-element in the static cutting region according to the following formula. 2,ij (t) and time delay coefficient τ ij The window function g of the axial infinitesimal element in the dynamic cutting region 2,ik (t) and time delay coefficient τ' ik :

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] Where "&" means "and", h ij h represents the instantaneous, undeformed chip thickness under low-frequency vibration of the j-th axial micro-element on the i-th tooth in the static cutting region. ik T represents the instantaneous, undeformed chip thickness under the low-frequency vibration of the k-th axial micro-element on the i-th cutting tooth in the dynamic cutting region. tp Let n be the tool's tooth-passing cycle, n' be the specific time delay coefficient when the time delay coefficient in the static cutting region is not 1, and n' be the specific time delay coefficient when the time delay coefficient in the static cutting region is not 1. Transition coefficient;

[0022] Step 5: Calculate the entry and exit angles for climb milling and conventional milling processes using the following formulas:

[0023] The formulas for calculating the entry and exit angles during climb milling are as follows:

[0024] φ st =arccos(a e / R-1)

[0025] φ ex =π

[0026] The formulas for calculating the entry and exit angles during the reverse milling process are as follows:

[0027] φ st =0

[0028] φ ex =arccos(1-a e / R)

[0029] Where φ st φ is the cutting angle of the cutting teeth. ex To cut the angle with the cutting teeth, a e The radial depth of cut is R, where R is the milling cutter radius.

[0030] Step 6: For the axial micro-elements in the static cutting region and the dynamic cutting region, calculate the window function corresponding to each cutting edge micro-element according to the following formula:

[0031]

[0032]

[0033] Where g 1,ij Let φ be the window function corresponding to the j-th axial infinitesimal element on the i-th tooth in the static cutting region. ij (t) represents the cutting angle corresponding to the j-th axial micro-element on the i-th tooth in the static cutting region, and g 1,ik Let φ be the window function corresponding to the k-th axial infinitesimal element on the i-th tooth in the dynamic cutting region. ik (t) represents the cutting angle corresponding to the k-th axial micro-element on the i-th cutting tooth in the dynamic cutting region;

[0034] Step 7: Calculate the dynamic milling force F in the feed direction of the robotic arm milling operation under the action of radial low-frequency vibration and axial low-frequency vibration according to the following formula. x (t), dynamic milling force F in the feed normal direction y (t) and dynamic milling force F in the tool axis direction z (t):

[0035]

[0036] G 1,ij (t)=g 1,ij (t)g 2,ij (t)

[0037] G 2,ik (t)=g 1,ik (t)g 3,ik (t)

[0038] Among them G 1,ij (t) is the static cutting region window function corresponding to the j-th axial infinitesimal element on the i-th cutting tooth, G 2,ik (t) is the dynamic cutting region window function corresponding to the k-th axial infinitesimal element on the i-th cutter tooth, A(φ ij (t) is the shear force direction factor matrix in the static cutting region, B(φ)ij (t) is the direction factor matrix of the damping force in the dynamic cutting region, A(φ) ik (t) is the shear force direction factor matrix in the static cutting region, B(φ) ik (t) is the direction factor matrix of the damping additional force in the dynamic cutting region, C pd,ij (t) represents the process damping coefficient in the static cutting region, C pd,ik (t) represents the damping coefficient term in the dynamic cutting region, w s,j w represents the axial micro-element height of the static cutting region. d,k Let be the axial infinitesimal height of the dynamic cutting region, and let x(t), y(t), and z(t) be the dynamic displacements in the feed direction, feed normal direction, and tool axis direction at time t, respectively. These represent the dynamic velocities at time t in the feed direction, feed normal direction, and tool axis direction, respectively; N is the number of teeth; and M is the number of teeth. s M represents the total number of infinitesimal elements in the static cutting region. dd This represents the total number of micro-elements in the dynamic cutting region.

[0039] Step 8: Establish a multi-delay dynamic milling force model according to the following formula, classify the axial micro-elements with the same time delay, and normalize the dynamic cutting region and the static cutting region:

[0040]

[0041]

[0042]

[0043]

[0044] in The time-term shearing coefficient matrix, The time-term process damping coefficient matrix, The time-delay term shearing coefficient matrix, η is the time delay coefficient; η is the time delay threshold, which is expressed by the following formula as the number of tooth cycles within one low-frequency vibration cycle:

[0045]

[0046] in For the "round up" calculation, T rv For low-frequency vibration period, T tp This is the tooth cycle;

[0047] Step 9: Establish the control equations for the milling dynamics of the tool-workpiece system based on the low-frequency vibration of the milling structure of the weak-stiffness configuration robotic arm according to the following formula:

[0048]

[0049] Where M, C, and K are the mass matrix, stiffness matrix, and damping matrix, respectively, which are 3×3 matrices, and q(t) = [x(t)y(t)z(t)]. T For dynamic displacement;

[0050] The stability of the equations is solved by using the multi-time-delay fully discretized method, resulting in a lobe diagram of milling stability considering the weak stiffness configuration of the robotic arm.

[0051] A computer system is characterized by comprising: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to implement the method described above.

[0052] A computer-readable storage medium is characterized by storing computer-executable instructions, which, when executed, are used to implement the above-described method.

[0053] Beneficial effects

[0054] This invention provides a method for predicting the stability of robotic arm milling. The method first divides the cutting area into axial regions; then, it breaks down the static and dynamic cutting areas into infinitesimal elements at unequal distances, calculating the window function and time delay coefficient required for judging the contact between the tool element and the workpiece; finally, it uses the obtained dynamic cutting force matrix and the stability solution of the multi-time-delay dynamic model of the milling process to complete the prediction of robotic arm milling stability. Compared with the given literature, this invention simultaneously considers the contact relationship between the tool and workpiece caused by axial low-frequency vibration and radial low-frequency vibration to establish a stability prediction method. This can solve the problem of the influence of severe low-frequency vibration on the machining process in robotic arm milling, and improve the prediction accuracy of the lobe diagram of robotic arm milling stability under weak stiffness configurations. Attached Figure Description

[0055] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts.

[0056] Figure 1 These are the dynamic cutting area and the static cutting area of ​​this invention.

[0057] Figure 2 This invention involves the axially non-equidistant micro-element discretization of the dynamic cutting region and the static cutting region.

[0058] Figure 3 These are the joint angles of the robotic arm used in the experiments of this invention. In the figure, "+" indicates the positive direction of the angle, and "-" indicates the negative direction of the angle.

[0059] Figure 4 This invention presents stability lobe diagrams considering low-frequency vibrations of radial and axial structures, a comparison of stability lobe diagrams considering only low-frequency vibrations of radial structures and stability lobe diagrams of traditional two-degree-of-freedom models, and experimental results. Detailed Implementation

[0060] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0061] This invention provides a method for predicting the stability of a robotic arm in milling. The method first divides the cutting region into a static cutting region unaffected by axial vibration and a dynamic cutting region affected by axial vibration. Then, it discretizes the two regions using unequally spaced axial elements, calculating the window function and time delay coefficient of the axial cutting elements in both the static and dynamic cutting regions. Finally, it combines the dynamic cutting force matrix and process damping matrix of each axial tool element in both the static and dynamic cutting regions to establish the control equations for the robotic arm's milling dynamics in the tool-workpiece system. By solving these equations for stability, a stability lobe diagram considering the weak stiffness configuration of the robotic arm can be plotted.

[0062] Expected technical effect: The dynamic modeling method for milling process of robot arm with weak stiffness configuration provided by the present invention can take into account the displacement changes caused by low-frequency vibration of tool-workpiece in both radial and axial directions, and predict the stability of the milling process of robot arm under weak stiffness configuration.

[0063] Example 1:

[0064] The experiment selected a flat-bottomed carbide end mill with 4 teeth (N=4), a radius (R=6 mm), and a helix angle (β=30 degrees) to be clamped on the end-effector of the robotic arm milling system and used climb milling for the cutting process. The radial cutting depth was a. e A milling test was conducted on an aluminum alloy block of 7075 with cutting parameters of 3 mm and feed per tooth c = 0.04 mm / tooth.

[0065] See attached document Figure 3 Use the robot arm posture with this joint angle rotation direction for milling. "+" and "-" indicate the positive and negative of the joint rotation direction. The angles of each joint are set according to the table below.

[0066] Table 1 Joint angles of the robotic arm used in the experiment

[0067]

[0068] Step 1: Determine the material physical parameter φ by referring to the cutting data published in the right-angle cutting parameter library. n ,β n and τ s Substitute the end mill's geometric parameters and the material's physical parameters into the following formula to calculate the cutting force coefficient:

[0069]

[0070]

[0071]

[0072] The radial cutting force coefficient K was calibrated. r = 237.6 N / mm², tangential cutting force coefficient K t = 979.9 N / mm², axial cutting force coefficient K a = 351.8 N / mm².

[0073] Step 2: Modal testing of the tool and workpiece is performed using the standard hammer impact method. A large hammer is used to obtain the structural modes, while a standard hammer is used to obtain the tool modes. The natural frequency matrix ω of the tool and robotic arm structure is obtained. n,c ω n,str Damping ratio matrix ζ c ζ str and modal mass matrix m c m str Where ω n,c Let ω be the natural frequency matrix of the cutting tool. n,str Let ζ be the natural frequency matrix of the robotic arm structure. c Let ζ be the tool damping ratio matrix. str Let m be the damping ratio matrix of the robotic arm structure. c Let m be the tool quality matrix. str The damping ratio matrix of the robotic arm structure is shown in Table 2. The structural modal parameters, obtained through hammer impact tests, are shown in Table 3. The structural modal directions include cross-coupling terms; A-direction - B-direction indicates that a hammer is applied to direction A, and accelerometer data is collected in direction B. The tool axis direction modes are predominantly spindle modes.

[0074] Table 2 Modal parameters of structural tests

[0075]

[0076]

[0077] Table 3 Modal parameters of the cutting tool test

[0078]

[0079] Step 3, refer to the appendix Figure 1 The workpiece cutting area is divided into a static cutting area and a dynamic cutting area according to the following formula, and then dispersed at unequal distances:

[0080]

[0081] Where w(t) is the actual axial cutting depth, w s w represents the axial cutting depth in the static cutting region. d (t) represents the axial cutting depth in the dynamic cutting region, w set For the set axial cutting depth, A ap A represents the maximum absolute amplitude of the axial low-frequency vibration. lv This represents the difference between the maximum and minimum axial cutting depths caused by low-frequency vibration. The displacement generated by the axial low-frequency vibration was obtained using the Newmark-β integral method proposed in the reference "NMNewmark, E. Mechanic, A Method of Computation for Structural Dynamics, 1959, p. 85."

[0082] Step 4, refer to the appendix Figure 2 The modeling of the static and dynamic cutting regions is analyzed, and the window function g of the axial infinitesimal element in the static cutting region is calculated according to the following formula. 2,ij (t) and time delay coefficient τ ij The window function g of the axial infinitesimal element in the dynamic cutting region 2,ik (t) and time delay coefficient τ i ' k .

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] The '&' symbol represents "and". ij h represents the instantaneous, undeformed chip thickness under low-frequency vibration of the j-th axial micro-element on the i-th tooth in the static cutting region. ik T represents the instantaneous, undeformed chip thickness under the low-frequency vibration of the k-th axial micro-element on the i-th cutting tooth in the dynamic cutting region.tp Let n be the tool's tooth-passing cycle, n' be the specific time delay coefficient when the time delay coefficient in the static cutting region is not 1, and n' be the specific time delay coefficient when the time delay coefficient in the static cutting region is not 1. Transition coefficients for ease of calculation.

[0089] Step 5: Calculate the entry and exit angles for climb milling and conventional milling processes using the following formulas:

[0090] 1. Formulas for calculating the entry and exit angles during climb milling:

[0091] φ st =arccos(a e / R-1)

[0092] φ ex =π

[0093] 2. Formulas for calculating the entry and exit angles during reverse milling:

[0094] φ st =0

[0095] φ ex =arccos(1-a e / R)

[0096] Where φ st φ is the cutting angle of the cutting teeth. ex To cut the angle with the cutting teeth, a e R represents the radial depth of cut, and R is the radius of the milling cutter.

[0097] For the climb milling process, φ st =arccos(a e / R-1)=arccos(3 / 6-1),φ ex =π

[0098] Step Six: For the axial micro-elements in the static cutting region and the dynamic cutting region, calculate the window function corresponding to each cutting edge micro-element according to the following formula:

[0099]

[0100]

[0101] Where g 1,ij Let φ be the window function corresponding to the j-th axial infinitesimal element on the i-th tooth in the static cutting region. ij (t) represents the cutting angle corresponding to the j-th axial micro-element on the i-th tooth in the static cutting region, and g 1,ik Let φ be the window function corresponding to the k-th axial infinitesimal element on the i-th tooth in the dynamic cutting region. ik(t) represents the cutting angle corresponding to the k-th axial micro-element on the i-th cutting tooth in the dynamic cutting region.

[0102] Step 7: Calculate the dynamic milling force F in the feed direction of the robotic arm milling operation under the action of radial low-frequency vibration and axial low-frequency vibration according to the following formula. x (t), dynamic milling force F in the feed normal direction y (t) and dynamic milling force F in the tool axis direction z (t):

[0103]

[0104] G 1,ij (t)=g 1,ij (t)g 2,ij (t)

[0105] G 2,ik (t)=g 1,ik (t)g 3,ik (t)

[0106] Among them G 1,ij (t) is the static cutting region window function corresponding to the j-th axial infinitesimal element on the i-th cutting tooth, G 2,ik (t) is the dynamic cutting region window function corresponding to the k-th axial infinitesimal element on the i-th cutter tooth, A(φ ij (t) is the shear force direction factor matrix in the static cutting region, B(φ) ij (t) is the direction factor matrix of the damping force in the dynamic cutting region, A(φ) ik (t) is the shear force direction factor matrix in the static cutting region, B(φ) ik (t) is the direction factor matrix of the damping additional force in the dynamic cutting region, C pd,ij (t) represents the process damping coefficient in the static cutting region, C pd,ik (t) represents the damping coefficient term in the dynamic cutting region, w s,j w represents the axial micro-element height of the static cutting region. d,k Let be the axial infinitesimal height of the dynamic cutting region, and let x(t), y(t), and z(t) be the dynamic displacements in the feed direction, feed normal direction, and tool axis direction at time t, respectively. These represent the dynamic velocities at time t in the feed direction, feed normal direction, and tool axis direction, respectively; N is the number of teeth; and M is the number of teeth. s M represents the total number of infinitesimal elements in the static cutting region. dd This represents the total number of micro-elements in the dynamic cutting region.

[0107] Step 8: Establish a multi-delay dynamic milling force model according to the following formula, classify the axial micro-elements with the same time delay, and normalize the dynamic cutting region and the static cutting region:

[0108]

[0109]

[0110]

[0111]

[0112] in The time-term shearing coefficient matrix, The time-term process damping coefficient matrix, The time-delay term shearing coefficient matrix, η is the time delay coefficient. η is the time delay threshold, expressed by the following formula as the number of tooth cycles within one low-frequency vibration cycle:

[0113]

[0114] in For the "round up" calculation, T rv For low-frequency vibration period, T tp This is the tooth cycle.

[0115] Step 9: Establish the control equations for the milling dynamics of the tool-workpiece system based on the low-frequency vibration of the low-stiffness configuration robotic arm milling structure according to the following formula:

[0116]

[0117] Where M, C, and K are the mass matrix, stiffness matrix, and damping matrix, respectively, which are 3×3 matrices, and q(t) = [x(t)y(t)z(t)]. T This is dynamic displacement.

[0118] The stability of the equations is solved using a multi-time-delay fully discretized method, resulting in a stability lobe diagram considering the weak stiffness configuration of the robotic arm. (See attached...) Figure 4 It can be seen that the lobe diagram of the robot arm milling stability obtained by using the dynamic model of the present invention that considers the weak stiffness configuration of the robot arm can better match the experimental results compared with the traditional two-degree-of-freedom milling model and the milling stability prediction method that only considers radial low-frequency vibration, thus proving the effectiveness of the proposed robot arm milling stability prediction method.

[0119] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in the present invention, and such modifications or substitutions should all be covered within the scope of protection of the present invention.

Claims

1. A method for predicting the stability of a robotic arm during milling, characterized in that... The steps are as follows: Step 1: Calibrate the tangential, radial, and axial cutting force coefficients through milling experiments; Step 2: Modal impact tests were conducted on the tool and robotic arm structure using the standard force hammer impact method to obtain the natural frequency matrix ω of the tool and robotic arm structure. n,c ω n,str Damping ratio matrix ζ c ζ str and modal mass matrix m c m str， Where ω n,c Let ω be the natural frequency matrix of the cutting tool. n,str Let ζ be the natural frequency matrix of the robotic arm structure. c Let ζ be the tool damping ratio matrix. str Let m be the damping ratio matrix of the robotic arm structure. c Let m be the tool quality matrix. str The damping ratio matrix of the robotic arm structure; Step 3: Divide the workpiece cutting area into a static cutting area and a dynamic cutting area according to the following formula, and then distribute them at unequal distances: Where w(t) is the actual axial cutting depth, w s w represents the axial cutting depth in the static cutting region. d (t) represents the axial cutting depth in the dynamic cutting region, w set For the set axial cutting depth, A ap A represents the maximum absolute amplitude of the axial low-frequency vibration. lv This is the difference between the maximum and minimum axial cutting depths caused by low-frequency vibration. Step 4: Analyze the modeling of the static and dynamic cutting regions, and calculate the window function g of the axial micro-element in the static cutting region according to the following formula. 2,ij (t) and time delay coefficient τ ij The window function g of the axial infinitesimal element in the dynamic cutting region 2,ik (t) and time delay coefficient τ i ' k : Where "&" means "and", h ij h represents the instantaneous, undeformed chip thickness under low-frequency vibration of the j-th axial micro-element on the i-th tooth in the static cutting region. ik T represents the instantaneous, undeformed chip thickness under the low-frequency vibration of the k-th axial micro-element on the i-th cutting tooth in the dynamic cutting region. tp Let n be the tool's tooth-passing cycle, n' be the specific time delay coefficient when the time delay coefficient in the static cutting region is not 1, and n' be the specific time delay coefficient when the time delay coefficient in the static cutting region is not 1. Transition coefficient; Step 5: Calculate the entry and exit angles for climb milling and conventional milling processes using the following formulas: The formulas for calculating the entry and exit angles during climb milling are as follows: f st =arccos(a e / R-1) f ex =π The formulas for calculating the entry and exit angles during the reverse milling process are as follows: f st =0 f ex =arccos(1-a e / R) Where φ st φ is the cutting angle of the cutting teeth. ex To cut the angle with the cutting teeth, a e The radial depth of cut is R, where R is the milling cutter radius. Step 6: For the axial micro-elements in the static cutting region and the dynamic cutting region, calculate the window function corresponding to each cutting edge micro-element according to the following formula: Where g 1,ij Let φ be the window function corresponding to the j-th axial infinitesimal element on the i-th tooth in the static cutting region. ij (t) represents the cutting angle corresponding to the j-th axial micro-element on the i-th tooth in the static cutting region, and g 1,ik Let φ be the window function corresponding to the k-th axial infinitesimal element on the i-th tooth in the dynamic cutting region. ik (t) represents the cutting angle corresponding to the k-th axial micro-element on the i-th cutting tooth in the dynamic cutting region; Step 7: Calculate the dynamic milling force F in the feed direction of the robotic arm milling operation under the action of radial low-frequency vibration and axial low-frequency vibration according to the following formula. x (t), dynamic milling force F in the feed normal direction y (t) and dynamic milling force F in the tool axis direction z (t): G 1,ij (t)=g 1,ij (t)g 2,ij (t) G 2,ik (t)=g 1,ik (t)g 3,ik (t) Among them G 1,ij (t) is the static cutting region window function corresponding to the j-th axial infinitesimal element on the i-th cutter tooth, G 2,ik (t) is the dynamic cutting region window function corresponding to the k-th axial infinitesimal element on the i-th cutter tooth, A(φ ij (t) is the shear force direction factor matrix in the static cutting region, B(φ) ij (t) is the direction factor matrix of the damping force in the dynamic cutting region, A(φ) ik (t) is the shear force direction factor matrix in the static cutting region, B(φ) ik (t) is the direction factor matrix of the damping additional force in the dynamic cutting region, C pd,ij (t) represents the process damping coefficient in the static cutting region, C pd,ik (t) represents the damping coefficient term in the dynamic cutting region, w s,j w represents the axial micro-element height of the static cutting region. d,k Let be the axial infinitesimal height of the dynamic cutting region, and let x(t), y(t), and z(t) be the dynamic displacements in the feed direction, feed normal direction, and tool axis direction at time t, respectively. These represent the dynamic velocities at time t in the feed direction, feed normal direction, and tool axis direction, respectively; N is the number of teeth; and M is the number of teeth. s M represents the total number of infinitesimal elements in the static cutting region. dd This represents the total number of micro-elements in the dynamic cutting region. Step 8: Establish a multi-delay dynamic milling force model according to the following formula, classify the axial micro-elements with the same time delay, and normalize the dynamic cutting region and the static cutting region: in The time term shearing coefficient matrix, The time-term process damping coefficient matrix is... The time-delay term shearing coefficient matrix, η is the time delay coefficient; η is the time delay threshold, which is expressed by the following formula as the number of tooth cycles within one low-frequency vibration cycle: in For the "round up" calculation, T rv For low-frequency vibration period, T tp This is the tooth cycle; Step 9: Establish the control equations for the milling dynamics of the tool-workpiece system based on the low-frequency vibration of the low-stiffness configuration robotic arm milling structure according to the following formula: Where M, C, and K are the mass matrix, stiffness matrix, and damping matrix, respectively, which are 3×3 matrices, and q(t) = [x(t)y(t)z(t)]. T For dynamic displacement; The stability of the equations is solved by using the multi-time-delay fully discretized method, resulting in a lobe diagram of milling stability considering the weak stiffness configuration of the robotic arm.

2. A computer system, characterized in that... include: One or more processors, a computer-readable storage medium for storing one or more programs, wherein, when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to implement the method of claim 1.

3. A computer-readable storage medium, characterized in that... The device stores computer-executable instructions, which, when executed, are used to implement the method of claim 1.

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