Spiral bevel gear tooth surface error compensation method considering machine tool geometric error

Abstract

The invention discloses a spiral bevel gear tooth surface error compensation method considering machine tool geometric errors, which relates to the technical field of machining, and comprises the following steps: step 1, establishing an initial machine tool space error model, establishing a tooth surface equation, and establishing a tooth surface error model; establishing a mapping relation from a geometric error to a tooth surface error according to the initial space error model and the tooth surface equation; 2, compensating the tool pose error based on a Jacobian matrix method, and calculating a tooth surface error after compensating the tool pose error; and 3, calculating a compensation value based on a particle swarm algorithm, compensating the tooth surface error obtained in the step 2, and outputting the compensated tooth surface. The compensation efficiency and accuracy during machining of the spiral bevel gear machine tool can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical processing, and in particular to a spiral bevel gear tooth surface error compensation method taking into account machine tool geometric errors. Background Art

[0002] Spiral bevel gears are a vital component widely used in mechanical transmissions. Their advantages include strong load-bearing capacity, high transmission efficiency, and low noise, leading to their widespread application in aerospace, automotive, and marine applications. However, the complex geometry of spiral bevel gear tooth surfaces makes high-precision manufacturing a key challenge. Spiral bevel gear grinding machines are key manufacturing equipment for spiral bevel gears. Grinding machine accuracy directly impacts the machining accuracy of spiral bevel gear tooth surfaces. This accuracy is influenced by numerous error sources, including geometric, thermal, force-induced deformation, vibration, dynamic, and servo errors. Geometric error accounts for approximately 40% of the total error, a figure significantly higher for precision and ultra-precision machine tools. Furthermore, these errors, such as thermal, force-induced deformation, vibration, dynamic, and servo errors, can all be converted into spatial geometric errors. Therefore, machine tool geometric error is the most fundamental error source and carries a higher control priority. Due to the numerous geometric error terms and the complex machining principles of spiral bevel gears, existing compensation methods struggle to establish a mapping between geometric error and tooth surface error, resulting in low machining accuracy. Summary of the Invention

[0003] The present invention aims to solve at least one of the technical problems existing in the prior art. To this end, the present invention proposes a method for compensating spiral bevel gear tooth surface errors taking into account machine tool geometric errors, which can improve the compensation efficiency and accuracy during spiral bevel gear machine tool machining.

[0004] According to an embodiment of the first aspect of the present invention, a method for compensating the tooth surface error of a spiral bevel gear taking into account the geometric error of a machine tool includes the following steps: Step 1: Establishing an initial machine tool spatial error model, establishing a tooth surface equation, and establishing a mapping relationship from geometric error to tooth surface error from the initial spatial error model and the tooth surface equation; Step 2: Compensating for the tool posture error based on the Jacobian matrix method, and calculating the tooth surface error after compensating for the tool posture error; Step 3: Calculating a compensation value based on a particle swarm algorithm, compensating for the tooth surface error obtained in step 2, and outputting the compensated tooth surface.

[0005] According to an embodiment of the present invention, a method for compensating the tooth surface error of a spiral bevel gear taking into account the geometric error of a machine tool has at least the following beneficial effects: the output tooth surface error is compensated by combining the Jacobian matrix method and the particle swarm algorithm. The Jacobian matrix may quickly find sensitive parameters in the local linearization process, while the PSO can optimize these parameters globally to avoid falling into local optimality. This combination may be more efficient than a single method, especially when dealing with high-dimensional, nonlinear problems. In addition, combining the two methods may improve the accuracy and robustness of the compensation while reducing the consumption of computing resources. Compared with the previous compensation methods for general machine tools, the method of this embodiment focuses more on special machine tools for spiral bevel gears, has higher compensation efficiency and accuracy for spiral bevel gear machine tools, and provides an efficient digital solution for the precision manufacturing of spiral bevel gears.

[0006] According to some embodiments of the present invention, establishing an initial machine tool spatial error model includes the following steps: analyzing the machine tool kinematic chain and geometric error model, listing the transformation matrix, and calculating the homogeneous transformation matrix from the grinding wheel to the workpiece in combination with the geometric error model.

[0007] According to some embodiments of the present invention, establishing a tooth surface equation includes the following steps: establishing an equation of an initial spiral bevel gear tooth surface, discretizing the tooth surface, and obtaining the tooth surface equation.

[0008] According to some embodiments of the present invention, establishing an equation for the initial spiral bevel gear tooth surface includes the following steps: calculating the posture error of the grinding wheel based on the homogeneous transformation matrix from the grinding wheel to the workpiece, and obtaining the equation for the initial spiral bevel gear tooth surface based on the geometric shape of the grinding wheel and the machine tool motion chain.

[0009] According to some embodiments of the present invention, the tooth surface is discretized, including the following steps: rotating the points on the gear tooth surface around the axis and projecting them onto a plane passing through the axis of rotation, wherein the range of the tooth surface point projection is a polygonal area composed of points G, H, I, and J, uniformly selecting N×M sample points within the polygonal area composed of points G, H, I, and J, calculating the coordinates of points G, H, I, and J, and obtaining the coordinate equations of the points on the tooth surface axial section based on the coordinates of points G, H, I, and J, and calculating the tooth surface equation based on the coordinate equations of the points on the tooth surface axial section.

[0010] According to some embodiments of the present invention, tool posture errors caused by sensitive error terms are compensated based on the Jacobian matrix method, including the following steps: extracting the tool posture vector, calculating the Jacobian matrix using the partial differential method, calculating the pseudo-inverse matrix of the Jacobian matrix, and determining the compensation amount based on the established spatial error model and substituting the error terms.

[0011] According to some embodiments of the present invention, extracting a tool pose vector comprises the following steps: extracting three or four columns of a transformation matrix of the machine tool forward kinematics as the tool pose vector based on an initial machine tool spatial error model.

[0012] According to some embodiments of the present invention, calculating the compensation value based on the particle swarm algorithm includes the following steps: calculating the compensation value with the tooth surface error as the optimization target.

[0013] According to some embodiments of the present invention, the compensation value is corrected with the tooth surface error as the optimization target, including the following steps: calculating individual extreme values, calculating global extreme values based on the individual extreme values, and outputting the compensated tooth surface.

[0014] According to some embodiments of the present invention, outputting the compensated tooth surface includes the following steps: substituting the global extreme value into the tooth surface equation to obtain the compensated tooth surface.

[0015] Additional aspects and advantages of the present invention will be set forth in part in the description which follows and, in part, will be obvious from the description which follows, or may be learned by practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] The present invention will be further described below with reference to the accompanying drawings and embodiments, in which: Figure 1 A schematic diagram of a spiral bevel gear machine tool according to an embodiment of the present invention; Figure 2 A schematic diagram of a kinematic chain of a spiral bevel gear machine tool according to an embodiment of the present invention; Figure 3 A schematic diagram of the geometrical morphology of a grinding wheel according to an embodiment of the present invention; Figure 4 A schematic cross-sectional view of a grinding wheel according to an embodiment of the present invention; Figure 5 A schematic diagram of a tooth surface point projection according to an embodiment of the present invention; Figure 6 This is a schematic diagram of tooth surface point discretization according to an embodiment of the present invention; Figure 7 This is a schematic diagram of an embodiment of the present invention when not compensated; Figure 8 This is a schematic diagram of an embodiment of the present invention after using the Jacobian matrix method to compensate for tool posture errors; Figure 9 This is a schematic diagram of tooth surface error compensation using a particle swarm algorithm according to an embodiment of the present invention. DETAILED DESCRIPTION

[0017] The following describes embodiments of the present invention in detail. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended only to explain the present invention and are not to be construed as limiting the present invention.

[0018] In the description of the present invention, it should be understood that descriptions involving orientation, such as the orientation or positional relationship indicated by up, down, etc., are based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention.

[0019] In the description of the present invention, "a plurality" refers to more than two. The use of "first" or "second" is solely for the purpose of distinguishing technical features and should not be construed as indicating or implying relative importance, or implicitly indicating the number of the indicated technical features, or implicitly indicating the order of the indicated technical features.

[0020] In the description of the present invention, unless otherwise clearly defined, terms such as setting, installing, and connecting should be understood in a broad sense, and technicians in the relevant technical field can reasonably determine the specific meanings of the above terms in the present invention based on the specific content of the technical solution.

[0021] Reference Figures 1 to 9 As shown, a method for compensating spiral bevel gear tooth surface errors taking into account machine tool geometric errors according to one embodiment of the present invention includes the following specific steps: Step 1: Establish the initial machine tool spatial error model of the six-axis five-link spiral bevel gear grinding machine. In this embodiment, the six-axis five-link spiral bevel gear grinding machine is taken as an example to explain the compensation method in detail. Figure 1 As shown in the figure, the machine tool consists of three parallel axes: X-axis, Y-axis, and Z-axis, and three rotating axes: A-axis, B-axis, and C-axis. The C-axis is a high-precision spindle, and its geometric error can be ignored. Figure 2 As shown, for this machine tool, the kinematic chain can be divided into a grinding wheel chain and a spiral bevel gear chain. The grinding wheel chain is grinding wheel → z-axis → y-axis → x-axis → machine tool, while the spiral bevel gear chain is machine tool → b-axis → a-axis → gear. After the initial machine tool spatial error model is established, the tooth surface equation is established. The mapping relationship between geometric error and tooth surface error is established based on the initial spatial error model and the tooth surface equation. Step 2: Compensate for tool posture error based on the Jacobian matrix method. The Jacobian matrix is a mathematical tool often used by many scholars to solve the inverse kinematics of five-axis machine tools. The Jacobian matrix describes the relationship between joint velocity and end position and posture through linear approximation. Position and posture are referred to as posture below. Using the Jacobian matrix can quickly approximate the inverse kinematic solution, avoiding the difficulty of deriving complex analytical solutions. It has the advantage of fast calculation speed and is especially suitable for the nonlinear kinematic model of five-axis machine tools. Calculate the tooth surface error after compensating for the tool posture error; Step 3: Calculate the compensation value based on the particle swarm algorithm, compensate the tooth surface error obtained in step 2, and output the compensated tooth surface. Particle swarm algorithms are widely used in optimization problems due to their strong versatility and fast convergence rate. Combining the Jacobian matrix method with the particle swarm algorithm to compensate for the output tooth surface error has the advantages of fast calculation speed and high accuracy. Compared with previous compensation methods for general machine tools, the method of this embodiment focuses more on spiral bevel gear machine tools, and has higher compensation efficiency and accuracy for spiral bevel gear machine tools.

[0022] Reference Figures 1 to 9 As shown, it can be understood that establishing the initial machine tool spatial error model includes the following specific steps: analyzing the machine tool motion chain and geometric error model. In the six-axis five-linkage spiral bevel gear grinding machine, each axis has geometric errors due to installation deviations and manufacturing defects in the motion axis. Generally speaking, each linear axis and rotary axis has six position-dependent geometric errors (PDGEs). In addition, there are three perpendicularity errors between the three linear axes, and each of the two rotary axes has two position errors and angular errors, for a total of eight. These eleven items do not change with the position of the machine tool and are called position-independent geometric errors (PIGEs). Specifically, these errors are: X-axis displacement error in the xyz direction δ x (x), δ y (x), δ z (x); Angular error ε of the X-axis in the xyz direction x (x), ε y (x), ε z (x); Displacement error δ of the Y axis in the xyz direction x (y), δ y (y), δ z (y); Angular error ε of the Y axis in the xyz direction x (y), ε y (y), ε z (y); Displacement error δ of the Z axis in the xyz direction x (z), δ y (z), δ z (z); Angular error ε of the Z axis in the xyz direction x (z), ε y (z), ε z (z); Displacement error δ of axis A in xyz direction x (a), δ y (a), δ z (a); Angular error ε of the A axis in the xyz direction x (a), ε y (a), ε z (a); Displacement error δ of the B axis in the xyz direction x (b), δ y (b), δ z (b); Angular error ε of the B axis in the xyz direction x (b), ε y (b), ε z (b); The perpendicularity error between the X-axis and the Y-axis S xy ; The perpendicularity error between the Y and Z axes S yz ; The perpendicularity error S between the Z axis and the X axis xz ; Installation position error of A-axis in the y direction δ Ay ; Installation position error of A-axis in z direction δ Az ; Installation attitude error of A axis around y direction γ AY ; Installation attitude error β of A-axis around z direction AZ ; Installation position error of B axis in x direction δ Bx ; Installation position error of B axis in z direction δ Bz ; Installation attitude error of B axis around x direction γ BX ; Installation attitude error α of the B-axis around the z directionBZ .

[0023] Tables 1 and 2 list the above errors in tabular form: Table 1 PDGEs in gear grinding machines

[0024] Table 2 PIGEs in gear grinding machines

[0025] according to Figure 2 The kinematic chain, Table 1 and Table 2, write out the transformation matrix:

[0026] Combined with the geometric error model, the homogeneous transformation matrix from the grinding wheel to the workpiece is calculated, which is specifically expressed as: (1).

[0027] Among them, x, y, z, a, b are the displacements of each axis, and the vector E=[e1,e2,…,e 41 ] T Indicates 41 geometric errors.

[0028] where δ = [δ x ,δ y ,δ z ] represents the position error of the grinding wheel in the xyz directions in the workpiece coordinate system, ε=[ε x ,ε y ,ε z ] represents the posture error of the grinding wheel in the xyz directions in the workpiece coordinate system.

[0029] Reference Figures 1 to 9 As shown, it can be understood that establishing the tooth surface equation includes the following steps: establishing the equation of the initial spiral bevel gear tooth surface, discretizing the tooth surface, and obtaining the tooth surface equation. Since the equation of the initial spiral bevel gear tooth surface is relatively complex and has no explicit solution, in order to conveniently obtain the tooth surface in practical applications, it is necessary to discretize the tooth surface to obtain a tooth surface equation with convenient calculation characteristics.

[0030] Reference Figures 1 to 9 As shown, it can be understood that establishing the equation of the initial spiral bevel gear tooth surface includes the following steps: calculating the posture error of the grinding wheel according to the homogeneous transformation matrix from the grinding wheel to the workpiece, and from Equation 1 combined with Table 1 and Table 2, the posture error of the grinding wheel under the influence of all geometric error terms can be expressed as: (2).

[0031] Figure 3is the geometric topography of the grinding wheel. According to the geometric topography of the grinding wheel and the kinematic chain of the machine tool, the grinding wheel in motion can be expressed as formula (3): (3).

[0032] Among them, M AZ It represents the transformation matrix from the A-axis (the installation position of the spiral bevel gear) to the Z-axis (the installation position of the grinding wheel), φ is the motion parameter of the grinding wheel (determines the position of the grinding wheel), and h and θ are the geometric parameters of the grinding wheel (determine the contact point of the grinding wheel).

[0033] The speed of each point on the grinding wheel can be expressed as formula (4): (4).

[0034] According to the meshing principle, the points on the tooth surface satisfy formula (5): (5).

[0035] Substituting formula (4) into formula (5), we have: (6).

[0036] The point with height h on the grinding wheel axis can be expressed as formula (7): (7).

[0037] The velocity of the point at height h on the grinding wheel axis can be expressed as formula (8): (8).

[0038] Considering Equation (8) and Equation (6) comprehensively, we can obtain Equation (9): (9).

[0039] Combining equations (9) and (3), we can obtain equation (10) for the initial spiral bevel gear tooth surface: (10).

[0040] Reference Figures 1 to 9 As shown, it can be understood that the discretization of the tooth surface includes the following steps: Figure 4 As shown, where δ f , δ, δ a Represent the root cone angle, node cone angle, and face cone angle, respectively. a represents the pitch radius, b represents the face width. f , δ, δ a 、 R a 、After b is determined, the points on the gear tooth surface are rotated around the axis and projected onto the plane passing through the axis of rotation. The projection range of the tooth surface points is limited to the polygonal area formed by points G, H, I, and J. N×M sample points are uniformly taken within the polygon formed by points G, H, I, and J. R ij and Z ij Represents the coordinates of each sample point, where i and j represent the index numbers of the sample point in the tooth height and tooth width directions, respectively, where i=1,2,…,N; j=1,2,…M. For points G, H, I, and J, their coordinates are as follows: (11).

[0041] (12).

[0042] (13).

[0043] (14).

[0044] According to the coordinates of points G, H, I, and J, the coordinates of any point on the polygon formed by points G, H, I, and J are obtained, as follows: (15).

[0045] If the three-dimensional coordinates of the tooth surface point in the gear coordinate system are (x ij ,y ij ,z ij ), then it and R, Z satisfy formula (16): (16).

[0046] Then the tooth surface point coordinates can be obtained by the simultaneous equation (17).

[0047] Combined with the previously established spatial error model, by substituting Equation (1) into Equation (17), the mapping relationship from geometric error to tooth surface error can be established.

[0048] Reference Figures 1 to 9 As shown, it can be understood that compensating the tool posture error caused by the sensitive error term based on the Jacobian matrix method includes the following steps: The motion quantities of the five axes in the machine tool coordinate system are as follows: (18).

[0049] Extract tool pose vector, tool pose vector is used Indicates that the Jacobian matrix is calculated using the partial differential method, as shown in formula (20): (20).

[0050] Calculate the pseudo-inverse matrix of the Jacobian matrix, as shown in formula (21): (twenty one).

[0051] The spatial error model is established based on formula (2) and the error term is substituted.

[0052] (twenty two).

[0053] where δ = [δ x ,δ y ,δ z ] represents the position error of the grinding wheel in the xyz directions in the workpiece coordinate system, ε=[ε x ,ε y ,ε z ] represents the posture error of the grinding wheel in the xyz directions in the workpiece coordinate system, and E represents the error term.

[0054] Based on the Jacobian matrix, the compensation amount d is determined by formula (23): x ,d y ,d z ,d a and d b .

[0055] (twenty three).

[0056] Reference Figures 1 to 9 As shown, it can be understood that extracting the tool pose vector includes the following steps: According to the initial machine tool spatial error model, the machine tool forward kinematic transformation matrix is given in formula (1), and the third and fourth columns of the matrix are extracted as the tool pose vector. Tool pose vector The calculation formula is: (19).

[0057] Reference Figures 1 to 9 As shown, it can be understood that the calculation of the compensation value based on the particle swarm algorithm includes the following steps: the compensation value is calculated with the tooth surface error as the optimization target. The particle swarm algorithm is initialized as a group of random particles (random solution), and then the optimal solution is found through iteration. In each iteration, the particle updates itself by tracking two extreme values; the first is the optimal solution found by the particle itself, which is called the individual extreme value; the other extreme value is the optimal solution currently found by the entire population, which is the global extreme value. Assume that in a D-dimensional target search space, there are N particles forming a colony, where the position of the i-th particle at time t is represented as a D-dimensional vector, as shown in formula (24): (twenty four).

[0058] The velocity of the i-th particle is also a D-dimensional vector, as shown in formula (25): (25).

[0059] According to the above theory, the particle's velocity and position update formula at time t+1 is as follows: (26).

[0060] (27).

[0061] Where r1 and r2 are random numbers between (0, 1), and c1 and c2 represent learning factors.

[0062] Reference Figures 1 to 9 As shown, it can be understood that the compensation value is corrected with the tooth surface error as the optimization target, including the following steps: calculating the individual extreme value, calculating the global extreme value based on the individual extreme value, the optimal position searched by the i-th particle so far is called the individual extreme value, and the calculation formula of the individual extreme value is: (28).

[0063] The optimal position searched by the entire particle swarm so far is the global extreme value, and the calculation formula of the global extreme value is: (29).

[0064] According to the calculated global extreme value, the compensated tooth surface is output.

[0065] Reference Figure 6 As shown, it can be understood that outputting the compensated tooth surface includes the following steps: after searching for the optimal solution, substituting the solution back into the tooth surface equation, that is, substituting it into formula (17), the compensated tooth surface can be obtained.

[0066] The embodiments of the present invention are described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made within the knowledge of ordinary technicians in the relevant technical field without departing from the scope of the present invention.

Claims

1. A method for compensating tooth surface errors of spiral bevel gears considering geometric errors of machine tools, characterized in that: The steps include: Step 1: Establish an initial machine tool spatial error model, establish a tooth surface equation, and establish a mapping relationship from geometric error to tooth surface error based on the initial spatial error model and the tooth surface equation; Step 2: Compensating the tool posture error based on the Jacobian matrix method, and calculating the tooth surface error after compensating the tool posture error; Step 3: Calculate the compensation value based on the particle swarm algorithm, compensate the tooth surface error obtained in step 2, and output the compensated tooth surface.

2. The spiral bevel gear tooth surface error compensation method considering machine tool geometric error according to claim 1, characterized in that: Establishing the initial machine tool spatial error model includes the following steps: analyzing the machine tool kinematic chain and geometric error model, listing the transformation matrix, and combining the geometric error model to calculate the homogeneous transformation matrix from the grinding wheel to the workpiece.

3. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 2, characterized in that: Establishing the tooth surface equation includes the following steps: establishing the equation of the initial spiral bevel gear tooth surface, discretizing the tooth surface, and obtaining the tooth surface equation.

4. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 3, characterized in that: Establishing the equation of the initial spiral bevel gear tooth surface includes the following steps: calculating the posture error of the grinding wheel according to the homogeneous transformation matrix from the grinding wheel to the workpiece, and obtaining the equation of the initial spiral bevel gear tooth surface according to the geometric shape of the grinding wheel and the machine tool kinematic chain.

5. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 4, characterized in that: The tooth surface is discretized, comprising the following steps: rotating the points on the gear tooth surface around the axis and projecting them onto a plane passing through the axis of rotation, wherein the range of the tooth surface point projection is a polygonal region formed by points G, H, I, and J, uniformly selecting N×M sample points within the polygonal region formed by points G, H, I, and J, calculating the coordinates of points G, H, I, and J, obtaining the coordinate equations of the points on the tooth surface axial section based on the coordinates of points G, H, I, and J, and calculating the tooth surface equation based on the coordinate equations of the points on the tooth surface axial section.

6. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 1, characterized in that: The tool posture error caused by sensitive error terms is compensated based on the Jacobian matrix method, which includes the following steps: extracting the tool posture vector, calculating the Jacobian matrix using the partial differential method, calculating the pseudo-inverse matrix of the Jacobian matrix, and determining the compensation amount based on the established spatial error model and substituting the error terms.

7. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 6, characterized in that: Extracting the tool pose vector includes the following steps: extracting the third and fourth columns of the transformation matrix of the machine tool forward kinematics as the tool pose vector according to the initial machine tool spatial error model.

8. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 1, characterized in that: Calculating the compensation value based on the particle swarm algorithm includes the following steps: calculating the compensation value with the tooth surface error as the optimization target.

9. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 8, characterized in that: The compensation value is corrected with the tooth surface error as the optimization target, including the following steps: calculating individual extreme values, calculating global extreme values based on the individual extreme values, and outputting the compensated tooth surface.

10. The method for compensating spiral bevel gear tooth surface errors considering machine tool geometric errors according to claim 9, characterized in that: Outputting the compensated tooth surface includes the following steps: substituting the global extreme value into the tooth surface equation to obtain the compensated tooth surface.

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