Process parameter synchronous optimization method for multi-axis machining of ball-end mill

Abstract

The process parameter synchronous optimization method of the ball end mill of the application solves the problems of incomplete process parameter optimization, difficulty in balancing precision and efficiency in the existing robot ball end mill multi-axis machining, and belongs to the field of robot milling machining. The application includes constructing a robot ball end mill multi-axis machining system model, calculating the force-induced displacement error of the contact point along the normal direction of the machining surface under any process parameter; establishing a geometric model of the tool-workpiece engagement area, calculating the area volume enclosed by the boundary curve of the tool-workpiece engagement area when milling at any robot posture; based on and, constructing a precision-efficiency comprehensive performance evaluation index; establishing a multi-domain constraint set; under the constraint of, taking the minimization of the precision-efficiency comprehensive performance evaluation index as the objective function, solving the optimal process parameter combination. The application realizes the synchronous optimization of multiple process parameters, and improves the robot machining precision and efficiency.

Description

Technical Field

[0001] This application relates to a method for synchronous optimization of process parameters in multi-axis machining of ball end mills, which belongs to the field of robotic milling. Background Technology

[0002] Industrial robots, due to their flexibility, large workspace, and multi-axis motion capabilities, have been widely used in the machining of complex structural components such as ship propellers and aerospace cabin structures. However, compared to traditional CNC machine tools, industrial robots have additional redundant degrees of freedom and lower structural rigidity, making them susceptible to cutting force excitation during the cutting process. This can lead to severe machining vibrations and tool tip deformation, resulting in a significant reduction in surface quality and accuracy. Currently, for localized machining scenarios of large and complex structural components, ball end mills are generally more suitable for this purpose.

[0003] In ball end milling, a near-zero cutting speed region exists near the tool tip. If the tool orientation is inappropriate, this region will primarily experience plowing and friction, easily leading to heat accumulation and force fluctuations. The local cutting speed increases as the radius increases from the tool tip outwards along the ball end. Compared to the fixed axial direction of traditional cylindrical end mills during milling, the axial direction of ball end mills is adjustable, typically characterized by the lead angle and tilt angle. An appropriate tilt angle can move the contact area away from the tool tip, improving effective cutting speed, chip removal, and surface quality. However, this also alters the tool-workpiece contact volume and the tooth entry / exit angle, thus simultaneously affecting the magnitude and direction of the cutting force and further complicating the contact relationship between the tool and workpiece. Furthermore, the system's response to spindle speed and feed rate is significantly correlated; even small changes in speed and orientation can cause significant differences in contact morphology, force spectrum, and stability region. Therefore, deformation errors and cutting efficiency during milling are highly sensitive to various factors such as feed direction, tool position, and redundancy angle. Synchronous optimization of these parameters becomes an important and challenging task. Furthermore, due to the low joint stiffness of industrial robots, the speed distribution and contact mechanism in the cutting zone amplify the impact on machining errors. Summary of the Invention

[0004] To address the issues of incomplete optimization of process parameters and difficulty in balancing accuracy and efficiency in multi-axis machining of ball end mills, this application provides a method for simultaneous optimization of process parameters in multi-axis machining of ball end mills.

[0005] This application discloses a method for synchronously optimizing process parameters in multi-axis machining of ball end mills, comprising:

[0006] S1. Construct a multi-axis machining system model for a robotic ball end mill and calculate the force-induced displacement error at the contact point along the normal direction of the machined surface under arbitrary process parameters. ;

[0007] S2. Establish a geometric model of the tool-workpiece meshing region and calculate the volume of the region enclosed by the boundary curves of the tool-workpiece meshing region during milling in any robot posture. ;

[0008] S3, Based on force-induced displacement error and region volume Construct a comprehensive performance evaluation index for accuracy and efficiency. :

[0009]

[0010] This serves as a reference value for force-induced displacement error. This is a volume reference value;

[0011] S4. Establish a multi-domain constraint set ;

[0012] S5. In multi-domain constraint sets Under the constraint of minimizing the comprehensive performance evaluation index of accuracy and efficiency. Given the objective function, solve for the optimal combination of process parameters. .

[0013] Preferably, the force-induced displacement error of the contact point along the normal direction of the machined surface is calculated under any process parameters. The methods include:

[0014] The multi-axis machining system model for robot ball end mills includes a robot kinematics model, a robot joint stiffness model, a milling force model, and a force-induced deformation prediction model.

[0015] Based on the milling force model, the cutting edge is discretized into infinitesimal elements. The tangential force, radial force, and axial force of each infinitesimal element are calculated. By integrating along the contact area of ​​the cutting edge, the triaxial force in the milling force coordinate system is obtained. , and After coordinate transformation, the three-axis milling force in the global coordinate system is obtained. :

[0016]

[0017] in, It is the rotation transformation matrix between the milling force coordinate system and the cutting contact coordinate system. It is the rotation transformation matrix between the tangential coordinate system and the global coordinate system;

[0018] The force-translational compliance submatrix is ​​obtained from the robot joint stiffness model. ;

[0019] Based on the triaxial milling force in the global coordinate system Sum of force-translation compliance submatrix The tool tip deformation in workpiece coordinates is obtained. :

[0020]

[0021] Given the rotation transformation matrix between the workpiece coordinate system and the global coordinate system, the force-induced deformation prediction model is as follows:

[0022]

[0023] Where n is the normal direction of the milled surface.

[0024] Preferably, the volume of the region enclosed by the boundary curve of the tool-workpiece meshing area during milling in any robot posture is... for:

[0025]

[0026] The tool-workpiece meshing area is enclosed by three curves, namely curves AB, BC, and CA. , , The boundary curve in the workpiece coordinate system CS W The triaxial parametric equations below, and These represent the upper and lower limits of the axial direction angle of the micro-element of the cutting edge. , R is the tool radius, and s is the tool path spacing. κ represents the surface normal cutting depth, and κ represents the axial position angle of the reference point on the cutting boundary.

[0027] As a preferred option, the reference value for force-induced displacement error for:

[0028]

[0029] in, The statistical value is the baseline test value. This represents the 95th percentile. A set of baseline poses to satisfy visibility, collision clearance and kinematic constraints.

[0030] As a preferred option, volume reference value for:

[0031]

[0032] This represents the intersection of the feasible region, which consists of the intersection of visibility / collision and stability maps, and robot / tool ​​constraints.

[0033] As a preferred option, the multi-domain constraint set is:

[0034]

[0035] in, For the lean angle yaw angle and feed direction angle The obtained global average sensitivity function, This represents the global average velocity sensitivity threshold;

[0036] For the machining program to be optimized, there are N tool positions. Given the tool position point and workpiece shape, calculate the i-th tool position point. The cutting tool and workpiece contact point , This indicates a conflict-free feasible space;

[0037] For the i-th cutter position The corresponding joint angle value of the robot pose, j=1,2,…6. , These are the upper and lower limits of the angle of the j-th joint of the robot, respectively; A specified factor less than 1 is used to determine the smoothness of the trajectory. Let j be the maximum angular velocity of the j-th joint. For the cutting tool from the contact point Move to the cutting point Time spent at the location;

[0038] Let Jacobi be the condition number. For the robot's Jacobian matrix, For the maximum Jacobian condition number constraint, Let be the dimension of the Jacobian matrix;

[0039] This represents the total cutting torque. The maximum limit of the total torque of the main spindle. For cutting power, This is the maximum limit of spindle power.

[0040] As a preferred option, the global average sensitivity function for:

[0041]

[0042] in, For the tool radius, This refers to the number of teeth on the milling cutter. The feed per tooth. This represents the number of all reference points in the tool-workpiece meshing area. Let be the axial position angle of the j-th point in the tool-workpiece meshing region;

[0043]

[0044] Let be the vector pointing from the tool center to the j-th point in the tool-workpiece meshing region. It is the unit direction vector of the tool axis defined by the tool attitude angle:

[0045]

[0046]

[0047] ;

[0048] in, The rotation matrix is ​​the rotation matrix from the contact coordinate system to the tool coordinate system. The three-axis rotation matrices are respectively composed of the tilt angle. Lean angle Feed direction angle get.

[0049] As a preferred option, in multi-domain constraint sets Under the constraint of minimizing the comprehensive performance evaluation index of accuracy and efficiency. Using the objective function, a step-by-step hybrid optimization strategy combining genetic algorithm and quadratic programming algorithm is employed to solve for the optimal combination of process parameters. ;

[0050] Among them, a genetic algorithm is used to perform a global coarse search to obtain a feasible optimal solution;

[0051] The final solution is obtained by performing a local fine search in the feasible optimal solution using a sequential quadratic programming algorithm.

[0052] As a preferred method, in which a genetic algorithm is used to perform a global coarse search to obtain a feasible optimal solution, the process parameters are combined. As chromosomes, perform a nearest-point feasible projection repair on individuals that do not satisfy the multi-domain constraint set:

[0053]

[0054] in, for The most recent individual, For the repaired individual;

[0055] The repaired individual is reassessed to determine whether it satisfies the multi-domain constraint set.

[0056] The fitness of an individual satisfying the multi-domain constraint set is:

[0057]

[0058] in, For fitness, basic fitness , Let u represent the u-th constraint in the multi-domain constraint set. Dimensional normalization for different constraints This is a coefficient that adaptively adjusts with the feasibility rate.

[0059] As a preferred method, in the approach of using a sequential quadratic programming algorithm to perform a local fine-grained search within feasible optimal solutions to obtain the final solution, the quadratic programming subproblem is constructed as follows:

[0060]

[0061] in, This is a search value among the feasible optimal solutions. Indicates the search direction; Represents the target gradient; Indicates the u-th constraint in gradient at;

[0062] It is a Lagrange function The Hessian BFGS approximation is used to characterize local curvature while maintaining positive definiteness; , Represents the u-th constraint in the multi-domain constraint set. Lagrange multipliers;

[0063] Target gradient:

[0064]

[0065] express gradient, express The gradient;

[0066] exist At this point, in solving the quadratic programming subproblem, the search direction is determined. ;

[0067] Calculated merit function exist Along the search direction directional derivative;

[0068]

[0069] in, Describes the merit function based on ℓ1. Indicates the penalty coefficient;

[0070] Based on the sufficient descent criterion of Wolfe's condition, a backtracking search is performed to determine the step size. This makes the step size Satisfying the merit function with respect to The sufficient descent condition;

[0071] according to Get the next search value .

[0072] The beneficial effect of this application is that the present invention constructs a comprehensive performance evaluation index for accuracy and efficiency. A multi-domain coupled constraint set is established, including velocity sensitivity constraints and various types of processing constraints; under the constraints of the multi-domain constraint set, the goal is to minimize... Using the objective function, a two-stage hybrid optimization strategy (GA-SQP) is employed to solve for the optimal combination of tool attitude angle, robot redundancy angle, and feed direction angle. This application achieves simultaneous optimization of multiple process parameters, improves machining accuracy and efficiency, enhances the practicality of the optimization scheme, and is suitable for high-efficiency and high-precision milling of complex structural parts. Attached Figure Description

[0073] Figure 1 A flowchart of the method in this application

[0074] Figure 2 The diagram shows the robot milling system and related coordinate systems. (a) is the robot coordinate system, (b) is the coordinate system on the tool and workpiece, and (c) is an angle diagram.

[0075] Figure 3 The tool-workpiece meshing area under different tool postures;

[0076] Figure 4 To optimize the flowchart step by step;

[0077] Figure 5 A schematic diagram showing the setup of the three sets of comparative experimental paths;

[0078] Figure 6 For the comparison of deformation error before and after optimization of the first group of experimental surfaces, ;

[0079] Figure 7 For the comparison of deformation error before and after optimization of the first group of experimental surfaces, ;

[0080] Figure 8 For the comparison of deformation error before and after optimization of the first group of experimental surfaces, change. Detailed Implementation

[0081] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of this application.

[0082] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other.

[0083] The present application will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the application.

[0084] The method for synchronously optimizing process parameters for multi-axis ball end mill machining in this embodiment includes:

[0085] Step 1: Construct a multi-axis machining system model for a robotic ball end mill and calculate the force-induced displacement error at the contact point along the normal direction of the machined surface under arbitrary process parameters. :

[0086] This implementation method is carried out on an industrial robot milling platform. To establish a unified reference system for each model, a definition is provided. Figure 2 The coordinate system shown:

[0087] : Global / robot base coordinate system, with its origin attached to the center of the robot's base, used to define various motion relationships in the robot's base coordinate system;

[0088] The robot flange coordinate system has its origin attached to the center of the robot flange and changes as the flange moves.

[0089] The workpiece coordinate system, with the intersection of the three bottom sides of the workpiece as its origin, is attached to the workpiece and used to define the machining surface and toolpath. The X coordinate system is set. W The projection of the axis onto the surface to be machined is X. W ′;

[0090] Tool coordinate system, fixed at the tool center origin, Z T The axis and the tool axis vector p coincide, X T Z-axis and flange coordinate system F The axes are parallel;

[0091] The tangential coordinate system takes the tool-workpiece contact point on the machined surface as its origin, n is the normal direction of the milled surface, f is parallel to the direction of the linear feed rate, and s is the cross product direction of n and f.

[0092] The milling force coordinate system is attached to the tool-workpiece contact point, and the tool vector p is Z. M The cross product of the tool axis vector p and the normal vector n of the surface to be machined is in the direction X. M axis.

[0093] The multi-axis machining system model for robot ball end mills includes a robot kinematics model, a robot joint stiffness model, a milling force model, and a force-induced deformation prediction model.

[0094] According to the elastic deformation assumption, the relationship between the force acting on EE and the generalized deformation is as follows:

[0095]

[0096] in, Let F be the deformation vector of the end effector, and F be the six-dimensional vector force applied to EE, including force and torque; C is the compliance matrix related to the Cartesian stiffness matrix K. In the actual milling process using the robot system, the robot is mainly affected by the cutting force on the end effector rather than the torque, and therefore the rotational deformation and torque generated by the cutting force can be ignored. Therefore:

[0097]

[0098] In the formula, △X t and △X r These are the translational deformation vector and rotational deformation vector of the end effector, respectively. , , and These represent the force-translational compliance submatrix, the force-rotation compliance submatrix, the torque-translational compliance submatrix, and the torque-rotation compliance submatrix, respectively. and These represent the force and torque vectors acting on EE, respectively, where:

[0099] ,

[0100] Since the direction of the milling force is defined in the milling force coordinate system {MCS}, while tool deformation is considered in the global coordinate system, the milling force in {MCS} needs to be converted into the milling force in the global coordinate system {GCS} before predicting the deformation of the robot end effector (EE). G F, its transformation is as follows:

[0101]

[0102] It is the rotation transformation matrix between the global coordinate system and the milling force coordinate system;

[0103] Based on the milling force model, the cutting edge is discretized into infinitesimal elements. The tangential force, radial force, and axial force of each infinitesimal element are calculated. By integrating along the contact area of ​​the cutting edge, the triaxial force in the milling force coordinate system is obtained. , and After coordinate transformation, the three-axis milling force in the global coordinate system is obtained. :

[0104]

[0105] In this application, only the influence of tool posture on milling force is considered in pose optimization; other milling parameters such as cutting speed, theoretical cutting width (toolpath spacing), and feed per tooth are considered fixed. Different feed directions will cause changes in the cutting force in the global coordinate system; therefore, the feed direction angle also needs to be considered in the parameters of robot pose optimization.

[0106] The flange end obtained from the above calculations is in the robot base coordinate system CS. G The deformation below is the force-induced deformation of the end effector EE. According to Hooke's law, the approximate deformation of the robot tool center point can be calculated.

[0107] Global Coordinate System CS G The following EE transformation is:

[0108]

[0109] Therefore, the tool tip deformation in workpiece coordinates for:

[0110]

[0111] This is the rotation transformation matrix between the workpiece coordinate system and the global coordinate system;

[0112] Since the surface error of the workpiece is determined by the component of the tool along the normal direction of the workpiece surface, the force-induced deformation in the normal direction of the surface to be machined is calculated, and the force-induced deformation prediction model is obtained as follows:

[0113]

[0114] ,in Forward lean angle, The roll angle is... The feed direction angle, This is a redundant angle.

[0115] The tool-workpiece meshing region (CWE) is a closed spherical geometry composed of three curves: AB, AC, and BC. AB is the intersection arc between the current tool sweep surface N and the plane perpendicular to the feed direction passing through the center of the tool hemisphere; AC is the intersection curve between the tool's rotating surface and the workpiece's surface to be machined H; and BC is the intersection curve between the tool's rotating surface and the previous toolpath transition surface M.

[0116] Step 2: Establish the geometric model of the tool-workpiece meshing region and calculate the volume of the region enclosed by the boundary curves of the tool-workpiece meshing region during milling in any robot posture. ;

[0117] Based on the concepts of infinitesimal element discretization and rigid body rotation transformation, this application considers the machining of general complex curved surfaces as a combination of machining of a series of tiny inclined planes. Furthermore, it argues that the tool contact area of ​​a ball end mill robot can be obtained by rotating the tool contact area of ​​a ball end mill vertically milling a horizontal surface. Specifically, when the robot is in its initial state, the tool coordinate system CS... T and workpiece coordinate system CS W Coincident, after giving the feed direction, along Z w After the axis rotates, it coincides with {CCS}. At this time, the feed direction angle β and the tool attitude angles γ and ε are all 0. The tool axis direction is perpendicular to the surface to be machined and coincides with the normal vector direction. At this time, curves AB, BC, and CA represent the initial state of the tool-workpiece meshing region. C The analytical models are as follows:

[0118]

[0119]

[0120]

[0121] In the formula, R is the radius of the ball end mill. This represents the parameters for climb milling and conventional milling (value is 1 for climb milling and -1 for conventional milling). κ represents the surface normal cutting depth, and s represents the tool path spacing. d and κ u These represent the upper and lower limits of the axial direction angle of the micro-element of the cutting edge. The radial position angle of the cutting edge element is used to represent the endpoint boundaries of curves AB and AC, respectively.

[0122]

[0123]

[0124]

[0125] The above analytical model lays the geometric foundation for calculating the CWE volume under different tool and feed directions. When the tool is in an arbitrary posture and the feed direction angle is not equal to 0, the analytical model of curves AB, BC, and CA can be obtained by rotational transformation of the closed spatial surface region enclosed by the above three boundary curves. The tool-workpiece meshing region enclosed by the three boundary curves when the tool is in three different postures in the tool coordinate system is as follows: Figure 3 As shown.

[0126] The aforementioned CWE region is moved from the tangential coordinate system CS C The rotation is transformed to the actual tool coordinate system. The composite rotation matrix is:

[0127]

[0128] Let a certain CWE point be in the feed coordinate system. Therefore, after the tool posture and feed direction change, its final expression in workpiece coordinates is:

[0129]

[0130] The above formula can be used to calculate the boundary curve of the tool-workpiece meshing area when the tool is milling in any robot posture. The volume of the enclosed area is calculated using the three-dimensional Green's theorem, and the calculation formula and specific algorithm are as follows:

[0131]

[0132] In the above formula, , , The boundary curve in the workpiece coordinate system CS W The triaxial parametric equations are given by CS. C The analytical model in the coordinate system is obtained by rigid body rotation C WR, where κ is the axial position angle of the reference point on the cutting boundary.

[0133] Step 3: Based on force-induced displacement error and region volume Construct a comprehensive performance evaluation index for accuracy and efficiency. ;

[0134] In robotic ball end milling, the selection of tool attitude angle and robot redundancy angle significantly affects machining accuracy (cutting force-induced normal error) and machining efficiency (material removal rate and process stability), and there is a significant dynamic coupling conflict between the two. To quantitatively evaluate this trade-off, this application proposes a novel dimensionless comprehensive performance evaluation index, namely the "Accuracy-Efficiency Index" (AEI). This index quantifies the "mass loss per unit equivalent cutting volume" by normalizing the force-induced deformation error and the material engagement volume, providing a theoretical basis for the synergistic optimization of accuracy, efficiency, and stability. The specific definition is as follows:

[0135]

[0136] The above indicator is a dimensionless comprehensive indicator, and the numerator of the indicator ( / FE ref The denominator represents the relative machining error deviation; while the denominator represents the relative machining error deviation. This reflects the normalized processing efficiency; the above formula eliminates dimensional differences, thus maintaining numerical balance and enhancing the index's adaptability to different processing conditions, while also increasing its sensitivity to accuracy loss. The index AEI compares "normalized accuracy loss" with "normalized volume." Here, FE is the force-induced displacement error at the contact point along the normal direction of the processed surface; while... This refers to the volume of the tool-workpiece meshing area. (Indicators) In the middle, the volume Mapped to length, making It becomes a dimensionless comprehensive index.

[0137] In order to It has a unified measurement standard and facilitates comparison, and reference values ​​FE are defined for different processing conditions. ref and V ref , representing the accuracy reference value and the volume reference value respectively, are introduced to normalize the indicators. The specific definitions and selection principles are as follows:

[0138] This application will FE ref Set it as the allowable value of the normal profile error required by the target machining task, or, when there is no explicit tolerance, take the statistical value of the baseline test for a robust estimate:

[0139]

[0140] Where Ω base A set of baseline attitudes to satisfy visibility (V-map), collision clearance, and kinematic constraints. FE is used. tol It allows AEI to directly reflect consistency with specifications; using the 95th percentile when there are no tolerances can avoid individual outliers dominating normalization.

[0141] At the same time, the volume reference value V ref Set as the maximum achievable CWE volume under the same operating envelope:

[0142]

[0143] in, The intersection of the feasible region, consisting of the intersection of the visibility / collision (V-map), the stability map, and the robot / tool ​​constraints, is as follows: This definition ensures that V ref As an upper bound for practically usable removal capabilities, As a normalization term from volume to length, it has physical meaning. In the experiments of this application, based on processing conditions and workpiece accuracy requirements, the following parameters were set... and The values ​​are 0.35mm and 33mm respectively. 3 .

[0144] Comprehensive performance evaluation indicators It directly reflects the balance between machining accuracy and cutting volume per unit time, and can uniformly evaluate the dynamic balance between accuracy, stability, and efficiency during machining. This indicator can quantify "mass loss per unit effective cutting volume," and its... A smaller value indicates better overall performance. While ensuring a larger effective cutting volume, it achieves lower force-induced errors, thus achieving the optimal balance between accuracy and efficiency, and better realizing the synergistic optimization of accuracy, efficiency, and stability. Therefore, It can serve as an important comprehensive performance objective function for optimizing tool attitude angles and redundancy angles in robotic milling.

[0145] The core significance of AEI lies in its multi-dimensional characteristic mapping and dynamic optimization capabilities for machining performance. Through an automatic trade-off mechanism in the form of ratios, this index effectively avoids the subjectivity of weight allocation in traditional single-objective machining, achieving coordinated decision-making for accuracy, efficiency, and stability. Normalized design provides cross-condition universality, adaptable to performance comparisons of different materials, tools, and robot systems. Furthermore, considering the dynamic characteristics of robot machining, The CWE volume suppresses the vibration risk caused by cutting force fluctuations and optimizes motion trajectory planning efficiency by linking it to material removal rate (MRR), significantly improving energy utilization in the machining process. In practical applications, this indicator provides a quantitative tool for process parameter selection, dynamic adaptive control, and equipment performance evaluation, and can significantly improve the synergistic optimization capability of "accuracy-efficiency-stability" in complex surface machining.

[0146] Step 4: Establish a multi-domain constraint set ;

[0147] (1) During ball end milling, the cutting speed in the tool-workpiece meshing area exhibits a highly nonlinear distribution, which has a significant impact on machining stability and surface quality. In order to improve the stability and machining adaptability of the optimized angle combination, this application introduces a speed sensitivity constraint function as an additional constraint condition in the optimization process to exclude tool posture and feed direction combinations that are abnormally sensitive to changes in cutting speed.

[0148] This application constructs a velocity proportional function at any tool contact point. This is used to characterize the coordination between feed rate and cutting speed; its derivative is used to construct the cutting speed sensitivity function. The value is then globally averaged within the tool-workpiece meshing region to measure the distribution ratio of low-speed cutting points in this region and assess the sensitivity of cutting forces to local speed changes during machining. This value is then used as a stability constraint in the optimization model, embedded within a multi-angle optimization framework to ensure that angle combinations meet process stability requirements. This constraint plays a crucial role in filtering unstable angle configurations, thereby promoting robustness and adaptive optimization of robotic ball end milling operations.

[0149] The specific definitions of the velocity proportional function and the local velocity sensitivity function at any tool contact point are as follows:

[0150]

[0151]

[0152] To comprehensively evaluate the stability of a given attitude angle and feed direction combination to velocity disturbances, this application extends the sensitivity function to the entire CWE region, constructing a global average sensitivity function, defined as:

[0153]

[0154] In the above formula, R is the tool radius. This refers to the number of teeth on the milling cutter. The feed per tooth. This represents the number of all reference points in the tool-workpiece meshing area. Let be the axial position angle of the j-th point in the tool-workpiece meshing region. This angle is determined by the tool posture, and its calculation expression is:

[0155]

[0156] in, The vector representing the center of the tool pointing to the j-th point in the CWE. It is the unit direction vector along the tool axis defined by the tool attitude angle, and the specific calculation formula is as follows:

[0157]

[0158]

[0159] The aforementioned speed proportionality function and speed sensitivity function vary with the tool attitude angle. These trends provide important insights for filtering unstable angle combinations and enhancing the robustness of robotic ball end milling processes.

[0160] The above function reflects the rate of change of the imbalance between feed rate and cutting speed in the tool-workpiece meshing region, and is an important indicator for local evaluation of machining stability. To avoid the angle combination being overly sensitive to local cutting speed changes and to ensure stable machining, the feasible region Ω is defined by the following constraints:

[0161]

[0162] (2) In an industrial robot machining environment, given the tool position point and workpiece shape, calculate the i-th tool position point. The cutting tool and workpiece contact point Arbitrary tool-workpiece contact point Tool attitude angle The maximum value range is For machining complex aerospace parts, in order to ensure interference-free operation between the tool and the workpiece, a conflict-free feasible space is configured using a discretized visual map method. This further limits the range of variation of the tool axis vector determined by the tool attitude angle (γ, ε). Therefore, when optimizing the tool axis vector direction, ensuring that it falls within the feasible region is crucial to avoid collisions, thus requiring the generation of each cutting contact point. The corresponding VMap.

[0163]

[0164]

[0165] In the formula, N is the total number of tool points. The quantity, as defined in this application, is the tool center point, which can be determined by the corresponding... and Unique definition.

[0166] This constraint ensures that all optimized tool orientations are physically feasible and collision-free in the robotic machining environment.

[0167] (3) Secondly, the robot's posture and position cannot exceed its range of motion. Therefore, each joint must satisfy the robot's kinematic constraints, and each joint angle must be subject to the joint angle limitation:

[0168]

[0169]

[0170] In the formula, This represents the matrix of angle values ​​for each joint of the robot corresponding to the contact point. and These represent the minimum and maximum reachable angles of the j-th joint, respectively. Kinematics solution for a robot; CL i The homogeneous transformation matrix from {FCS} to {WCS} corresponding to the point.

[0171] This constraint ensures that all robot configurations for evaluated postures are kinematically feasible and do not exceed the limitations of the robot's physical joints during processing.

[0172] (4) Furthermore, the robot's trajectory must be smooth when it moves, and the angle of each joint cannot change drastically. Therefore, it is necessary to impose a maximum angular velocity constraint on the rate of change of the joint angles:

[0173]

[0174] In the formula, For the i-th cutter position The j-th joint angle value corresponding to the robot pose. This represents the maximum angular velocity of each joint of the robot. This represents a specified factor less than 1, used to determine the smoothness of the trajectory. Let j be the maximum angular velocity of the j-th joint. For the cutting tool from the contact point Move to the cutting point Time spent at the location.

[0175] This constraint ensures the continuity and smoothness of joint movement, avoiding high-frequency fluctuations that could negatively impact machining quality and dynamic stability.

[0176] (5) During the robot's machining process, it is difficult to solve the inverse kinematics problem at singular points, and the joint rotation speed will suddenly increase significantly. Therefore, it is necessary to avoid the robot passing through singular points and to constrain the condition number of the Jacobian matrix:

[0177]

[0178] In the formula, Let Jacobi be the condition number. For the robot's Jacobian matrix, For the maximum Jacobian condition number constraint, Let be the dimension of the Jacobian matrix. It can be seen that... The range of values ​​is The larger the condition number, the closer it is to the singular location; the closer it is to 1, the farther the robot is from the singular region. The maximum condition number constraint is set.

[0179] By enforcing this constraint, the optimization process ensures that all candidate poses remain within a well-conditioned and numerically stable region of the robot's workspace.

[0180] In summary, a multi-domain constraint set can be defined, as shown in the following formula.

[0181]

[0182] In the above formula, Obtained from robot inverse kinematics, This represents the total cutting torque. The maximum limit of the total torque of the main spindle. For cutting power, This is the maximum limit of spindle power.

[0183] Step 5: In the multi-domain constraint set Under the constraint of minimizing the comprehensive performance evaluation index of accuracy and efficiency. Given the objective function, solve for the optimal combination of process parameters. .

[0184] This application proposes an optimization model for tool attitude angle and redundancy angle. The optimization model consists of the optimization object, constraints, and objective function. Using the comprehensive performance index (AEI) as the driving objective, and combining dynamic constraints with a multi-algorithm collaborative strategy, the system solves for the tool attitude angle and redundancy angle. and feed direction angle The optimal combination of process parameters is found. This method can coordinate and optimize tool orientation and robot configuration, thereby achieving high-precision and high-efficiency robotic ball end milling.

[0185] The decision vector and the optimization objective are shown in the following equations.

[0186]

[0187] The optimization objective is to minimize the comprehensive performance evaluation index AEI:

[0188]

[0189] This application employs a step-by-step hybrid optimization strategy of "global coarse search + local fine search," balancing the exploration capability of the global solution with the progress stability of local optima. This effectively obtains the optimal combination of tool attitude angle, redundancy angle, and feed direction angle. The overall flowchart is shown below. Figure 4 As shown.

[0190] In multi-domain constraint sets Under the constraint of minimizing the comprehensive performance evaluation index of accuracy and efficiency. Using the objective function, a step-by-step hybrid optimization strategy combining genetic algorithm (GA) and quadratic programming algorithm (SQP) is employed to solve for the optimal combination of process parameters. ;

[0191] Among them, a genetic algorithm (GA) is used to perform a global coarse search to obtain a feasible optimal solution;

[0192] The final solution is obtained by performing a local fine search in the feasible optimal solution using the Sequential Quadratic Programming (SQP) algorithm.

[0193] (1) Stage 1: Genetic Algorithm (GA) Global Coarse Search

[0194] Angle combinations are achieved using real-number encoding. Represented as chromosomes. The variable is generated using maximin-LHS sampling of size N within a given boundary value range. p The initial population is linearly scaled to the actual range of values ​​according to the upper and lower bounds of the variables. For individuals that do not meet the constraint set, a nearest-point feasible projection is performed to fix them.

[0195]

[0196] The repaired individual is reassessed to determine whether it satisfies the multi-domain constraint set.

[0197] Individuals satisfying the multi-domain constraint set enter fitness evaluation and evolution. Based on fitness; penalized fitness is used for ranking selection:

[0198]

[0199] in, It adaptively adjusts based on the feasibility rate (increasing if the feasibility rate is low). Dimensional normalization is used for different constraints. The evolutionary operator employs tournament selection, simulated binary crossover (SBX), and multi-point Gaussian mutation, iterating cyclically until convergence or the computational budget is exhausted.

[0200] The convergence criterion is: when the optimal individual In continuous Intra-generation improvement is less than the threshold or reaching the maximum algebra The search process terminates at this point. The top 5% of feasible solutions are then used as the initial values ​​for SQP local refinement. In the coarse search phase, this application uses the following parameters: population size... Crossover probability Probability of mutation , This stage aims to quickly locate high-quality solution regions and avoid the risk of getting trapped in local extrema.

[0201] (2) Stage 2: Sequential Quadratic Programming (SQP) Local Fine Search

[0202] Based on the feasible optimal solution obtained from GA, SQP is used for local fine-grained search to obtain the final solution. Let the decision vector be... At the current iteration point At this point, using a quadratic model to approximate the objective and a linearized approximation constraint, a quadratic programming (QP) subproblem is constructed and variable boundaries are applied:

[0203]

[0204] In the formula, Represents the combined step length; Represents the target gradient; Represents the u-th constraint. gradient at; It is a Lagrange function The Hessian (damped) BFGS approximation is used to characterize local curvature and remain positive definite.

[0205] The gradient of the objective function is calculated using the chain rule:

[0206]

[0207] in, Depend on The chain rule is used to obtain the derivative; It is the volume pair of the CWE closed body The complex step / numerical sensitivity is obtained. Constrained gradient. (include: Stability, V-map visibility / collision-free performance, joint upper and lower bounds and attitude continuity, Jacobian condition number, power / torque, etc. are calculated using the complex step method / numerical sensitivity robustness method; gradients are truncated or projected near feasible boundaries to avoid numerical instability.

[0208] To ensure global convergence and feasibility, the step size adopts a merit function. Wolfe conditional backtracking search adaptive determination:

[0209]

[0210] When linearization is not feasible, or line search leads to a serious default, first perform nearest-point feasible projection on the current point, and increase the penalty coefficient if necessary. Alternatively, a filter strategy can be adopted.

[0211] The termination criterion for this stage is set to stop refinement when any of the following conditions are met: ① (First-order optimality); ② (Maximum constraint violation minimum); ③ Number of iterations An AEI change rate of <0.5% is used as an auxiliary stopping condition. After convergence, IK / collision / continuity re-verification is performed, and the optimal quality solution is output. .

[0212] Case Analysis:

[0213] To verify the effectiveness of the proposed method for jointly optimizing tool attitude angle and feed direction, three machining paths were designed, and three sets of comparative experiments were conducted. The first two sets of paths had a feed direction angle of 90°. o and 0 o The first group of paths is a straight path, while the third group of paths is a curved path where the feed direction angle changes. Figure 5 As shown, each set of experiments was performed on the same workpiece and included two posture combinations before and after optimization.

[0214] A CR-SCAN RAPTOR 3D scanner was used to perform in-situ planar acquisition of the two cutting surfaces in each experiment. Then, the point cloud tool CloudCompare was used to collect the obtained point cloud data of the machined surfaces. Specifically, 100 points were uniformly sampled on each machined surface, and the vertical distance of each sampling point relative to the reference plane was measured. This yielded the deformation error values ​​of each sampling point before and after optimization. The error values ​​of the measured points on the machined surfaces before and after optimization for each experiment are shown below. Figures 6 to 8 As shown in Table 1, the statistical measures of the experimental results for each group are calculated, including: mean error, standard deviation, minimum error value, and maximum error value.

[0215]

[0216] The results above show that when the feed direction angle is 0... o and 90 oAfter optimization, the average errors of both experimental groups significantly decreased from 0.599 mm and 0.514 mm to 0.24 mm and 0.349 mm, representing reductions of 59.9% and 32.1%, respectively. The maximum errors decreased from 1.102 mm and 1.340 mm to 0.423 mm and 0.688 mm, and the standard deviation of the error distribution decreased from 0.231 and 0.267 to 0.091 and 0.189. The results show that the optimized attitude combination significantly improves machining accuracy and surface consistency, demonstrating superiority over traditional attitude settings. Further comparison of the optimized data shows that the second group outperforms the first group in both average error and error dispersion, indicating that the feed direction angle has a significant impact on error sensitivity. =90 o This approach is more conducive to fully leveraging the advantages of the optimization strategy in stability control. Both sets of results jointly verify the applicability and robustness of the AEI optimization strategy in this application under different path directions. The third set of experiments used a more complex curved path. After optimization, the average error decreased from 0.449 mm to 0.279 mm, a reduction of 37.9%, and the maximum error decreased from 0.821 mm to 0.552 mm, showing a good error suppression effect. However, its standard deviation increased from 0.101 to 0.233, indicating that although the AEI index effectively reduced the average error, insufficient attitude continuity and drastic changes in redundancy angles may lead to increased local error fluctuations, affecting the machining stability under complex paths.

[0217] While this application has been described with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of this application. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of this application as defined by the appended claims. It should be understood that different dependent claims and features described in this application can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for synchronously optimizing process parameters in multi-axis machining of ball end mills, characterized in that, include: S1. Construct a multi-axis machining system model for a robotic ball end mill and calculate the force-induced displacement error at the contact point along the normal direction of the machined surface under arbitrary process parameters. ; S2. Establish a geometric model of the tool-workpiece meshing region and calculate the volume of the region enclosed by the boundary curves of the tool-workpiece meshing region during milling in any robot posture. ; S3, Based on force-induced displacement error and region volume Construct a comprehensive performance evaluation index for accuracy and efficiency. : ； This serves as a reference value for force-induced displacement error. This is a volume reference value; S4. Establish a multi-domain constraint set ; S5. In multi-domain constraint sets Under the constraint of minimizing the comprehensive performance evaluation index of accuracy and efficiency. Given the objective function, solve for the optimal combination of process parameters. .

2. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 1, characterized in that, Calculate the force-induced displacement error of the contact point along the normal direction of the machined surface under arbitrary process parameters. The methods include: The multi-axis machining system model for robot ball end mills includes a robot kinematics model, a robot joint stiffness model, a milling force model, and a force-induced deformation prediction model. Based on the milling force model, the cutting edge is discretized into infinitesimal elements. The tangential force, radial force, and axial force of each infinitesimal element are calculated. By integrating along the contact area of ​​the cutting edge, the triaxial force in the milling force coordinate system is obtained. , and After coordinate transformation, the three-axis milling force in the global coordinate system is obtained. : ； in, It is the rotation transformation matrix between the milling force coordinate system and the cutting contact coordinate system. It is the rotation transformation matrix between the tangential coordinate system and the global coordinate system; The force-translational compliance submatrix is ​​obtained from the robot joint stiffness model. ; Based on the triaxial milling force in the global coordinate system Sum of force-translation compliance submatrix The tool tip deformation in workpiece coordinates is obtained. : ； Given the rotation transformation matrix between the workpiece coordinate system and the global coordinate system, the force-induced deformation prediction model is as follows: ； in, It is the normal direction of the milled surface.

3. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 1, characterized in that, Volume of the region enclosed by the boundary curve of the tool-workpiece meshing area during arbitrary orientation milling for: ； The tool-workpiece meshing area is enclosed by three curves, namely curves AB, BC, and CA. , , The boundary curve in the workpiece coordinate system CS W The triaxial parametric equations below, and These represent the upper and lower limits of the axial direction angle of the micro-element of the cutting edge. , R is the tool radius, and s is the tool path spacing. κ represents the surface normal cutting depth, and κ represents the axial position angle of the reference point on the cutting boundary.

4. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 1, characterized in that, Reference value of force-induced displacement error for: ； in, The statistical value is the baseline test value. This represents the 95th percentile. A set of baseline poses to satisfy visibility, collision clearance and kinematic constraints.

5. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 1, characterized in that, Volume reference value for: ； This represents the intersection of the feasible region, which consists of the intersection of visibility / collision and stability maps, and robot / tool ​​constraints.

6. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 1, characterized in that, The set of multi-domain constraints is: ； in, For the lean angle yaw angle and feed direction angle The obtained global average sensitivity function, This represents the global average velocity sensitivity threshold; For the machining program to be optimized, there are N tool positions. Given the tool position point and workpiece shape, calculate the i-th tool position point. The cutting tool and workpiece contact point , This indicates a conflict-free feasible space; For the i-th cutter position The corresponding joint angle value of the robot pose, j=1,2,…6. , These are the upper and lower limits of the angle of the j-th joint of the robot, respectively; A specified factor less than 1 is used to determine the smoothness of the trajectory. Let j be the maximum angular velocity of the j-th joint. For the cutting tool from the contact point Move to the cutting point Time spent at the location; Let Jacobi be the condition number. For the robot's Jacobian matrix, For the maximum Jacobian condition number constraint, Let be the dimension of the Jacobian matrix; This represents the total cutting torque. The maximum limit of the total torque of the main spindle. For cutting power, This is the maximum limit of spindle power.

7. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 6, characterized in that, Global average sensitivity function for: ； in, For the tool radius, This refers to the number of teeth on the milling cutter. The feed per tooth. This represents the number of all reference points in the tool-workpiece meshing area. Let be the axial position angle of the j-th point in the tool-workpiece meshing region; ； Let be the vector pointing from the tool center to the j-th point in the tool-workpiece meshing region. It is the unit direction vector of the tool axis defined by the tool attitude angle: ； ； ； in, Let R be the rotation matrix from the contact coordinate system to the tool coordinate system, and R be the tool radius. The three-axis rotation matrices are respectively determined by the tilt angle. Lean angle Feed direction angle get.

8. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 1, characterized in that, In multi-domain constraint sets Under the constraint of minimizing the comprehensive performance evaluation index of accuracy and efficiency. Using the objective function, a step-by-step hybrid optimization strategy combining genetic algorithm and quadratic programming algorithm is employed to solve for the optimal combination of process parameters. ; Among them, a genetic algorithm is used to perform a global coarse search to obtain a feasible optimal solution; The final solution is obtained by performing a local fine search in the feasible optimal solution using a sequential quadratic programming algorithm.

9. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 8, characterized in that, In the method of using a genetic algorithm for global coarse search to obtain feasible optimal solutions, process parameters are combined. As chromosomes, perform a nearest-point feasible projection repair on individuals that do not satisfy the multi-domain constraint set: ； in, for The most recent individual, For the repaired individual; The repaired individual is reassessed to determine whether it satisfies the multi-domain constraint set. The fitness of an individual satisfying the multi-domain constraint set is: ； in, For fitness, basic fitness , Let u represent the u-th constraint in the multi-domain constraint set. Dimensional normalization for different constraints This is a coefficient that adaptively adjusts with the feasibility rate.

10. The method for synchronous optimization of process parameters for multi-axis machining of ball end mills according to claim 8, characterized in that, In the method of using a sequential quadratic programming algorithm to perform a local fine-grained search on feasible optimal solutions to obtain the final solution, the quadratic programming subproblem is constructed as follows: ； in, This is a search value among the feasible optimal solutions. Indicates the search direction; Represents the target gradient; Indicates the u-th constraint in gradient at; It is a Lagrange function The Hessian BFGS approximation is used to characterize local curvature while maintaining positive definiteness; , Represents the u-th constraint in the multi-domain constraint set. Lagrange multipliers; Target gradient: ； express gradient, express The gradient; exist At this point, in solving the quadratic programming subproblem, the search direction is determined. ; Calculate the merit function exist Along the search direction directional derivative; ； in, Describes the merit function based on ℓ1. Indicates the penalty coefficient; Based on the sufficient descent criterion of Wolfe's condition, a backtracking search is performed to determine the step size. This makes the step size Satisfy the good value function The sufficient descent condition; according to Get the next search value .

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