CN106649996B - Consider the multi-axis milling tool axis modeling method of cutter bounce - Google Patents

Abstract

The invention proposes a multi-axis milling tool axis modeling method considering tool runout, provides a spindle motion model and a tool installation error model considering the radial error of the bearing inner ring, and establishes a local coordinate system to represent the components of the spindle system and their Then, the coordinate transformation method is used to obtain the tool axis motion model of the spindle motion error and the installation error of the tool holder and the collet. The invention solves the mathematical modeling problem of the tool axis motion model under the consideration of the influence of the spindle motion error and the tool installation error on the axis motion of the milling tool, and uses the particle swarm algorithm to calibrate the unknown parameters in the model. The model established by the invention can clearly describe the motion of the main shaft, and the proposed parameter calibration method has the characteristics of simple measurement process and fast data processing, and provides a theoretical basis for suppressing tool runout and improving machining accuracy, so that the model can be used in the machining process. Errors introduced by each component are analyzed.

Description

Modeling method of multi-axis milling tool axis considering tool runout

technical field

The invention belongs to the technical field of multi-axis numerical control machining, and relates to the research on the problem of modeling the axis motion of a milling tool considering the movement error of the machine tool spindle and the installation error of the tool, in particular to a multi-axis milling tool axis modeling method considering the tool runout.

Background technique

In CNC machining, tool runout will cause the instantaneous pose of the tool axis to deviate from the ideal state, change the relative positional relationship between the tool and the workpiece, and cause uneven cutting thickness of each tooth, which is prone to over-cut, under-cut and other phenomena. Tool runout cannot be completely avoided, which seriously restricts the further improvement of machining accuracy. In the case of tool runout, establishing a tool axis motion model has become one of the hotspots in current research. Due to the manufacturing and installation errors of the machine tool spindle, tool holder and tool, it is inevitable that the tool axis of rotation does not coincide with the tool axis during the milling process. Usually, several position parameters are used to define the relationship between the tool axis and the tool axis of rotation. relative position and describe the motion of the tool axis.

In the existing research, the machine tool spindle motion error and tool installation error are not effectively integrated as research objects. The research of Shanghai Jiaotong University simplified the tool runout into a model including parameters such as eccentric distance ρ, eccentric angle λ, inclination angle τ and torsion angle φ, and calibrated with experimental data. Although this method is easy to understand and describe the phenomenon of tool runout, it is not conducive to further clarifying the coupling effect and physical significance of tool clamping system manufacturing and installation errors on tool runout, and is not conducive to establishing a quantitative and accurate relationship between various error factors and tool runout .

In order to solve the problem of modeling the axis motion of the milling tool due to the motion error of the machine tool spindle and the tool installation error, the present invention uses the motion trajectory of the metal standard ball of the spindle dynamic error analyzer as the axis motion trajectory of the milling tool, and establishes the spindle motion error and the tool installation. The error model was used to analyze the influence of the axis movement of the milling tool, and the tool axis movement model was obtained, and the unknown parameters in the model were calibrated using the optimization algorithm. Through the experimental verification, it can be seen that the model established in this paper can clearly describe the motion of the main shaft. The proposed parameter calibration method has the characteristics of simple measurement process and fast data processing.

SUMMARY OF THE INVENTION

The invention provides a research method of the spindle motion error modeling and the tool installation error modeling. Considering the influence of the spindle motion error and tool installation error on the axis motion of the milling tool, the mathematical modeling problem of the tool axis motion model is solved, and the unknown parameters in the model are calibrated using the particle swarm algorithm. The model established by the invention can clearly describe the movement of the main shaft, and the proposed parameter calibration method has the characteristics of simple measurement process and fast data processing, and provides a theoretical basis for suppressing tool runout and improving machining accuracy, so that in the machining process The error introduced by each component can be analyzed according to the model.

The technical scheme of the present invention is:

The present invention firstly provides a spindle motion model and a tool installation error model considering the radial error of the bearing inner ring, establishes a local coordinate system to represent the components of the spindle system and their errors, and then uses a coordinate transformation method to obtain the spindle motion error, tool The tool axis kinematic model for the installation error between the shank and the collet.

The method for modeling the axis of a multi-axis milling tool considering tool runout is characterized by comprising the following steps:

Step 1: Establish a coordinate system CS1 at the installation position of the front end bearing of the multi-axis vertical milling machining center, where the center of the front end bearing hole is the coordinate origin O ₁ of CS1, and the X ₁ , Y ₁ , and Z ₁ axes of CS1 are respectively related to the machine tool coordinate system The X ₀ , Y ₀ , Z ₀ axes of CS0 are parallel;

In the coordinate system CS1, the parameter equation for establishing the motion trajectory _of the spindle axis A1 at the rear bearing position is as follows:

_The parametric equation for establishing the motion trajectory of the spindle axis A2 at the position of the front end bearing is as follows

Among them, h is the distance between the installation position of the front end bearing and the rear end bearing, ω is the spindle speed; a ₁ , a ₂ , b ₁ and b ₂ are the radial parameters describing the motion of the spindle, and θ ₁ and θ ₂ are the angles describing the motion of the spindle parameter;

Establish a coordinate system CS2, take the center of the bottom end of the machine tool spindle as the coordinate origin O ₂ of CS2, the axis direction of the spindle is the Z ₂ axis direction of CS2, the plane P ₂ of the bottom end of the spindle is the X ₂ Y ₂ plane of CS2, O ₁ A ₂ and the A ₁ A ₂ The intersection of the plane P ₁ and the plane P ₂ where the two straight lines are located is the X ₂ axis of CS2, and the transformation matrix M ₂₁ from the coordinate system CS2 to the coordinate system CS1 is obtained as

where a _2,x , b _2,x , c _2,x are the cosine of the direction of the X ₂ axis of the CS2 coordinate system in the coordinate system CS1, a _2,y , b _2,y , c _2,y are the CS2 coordinate system The direction cosine of the Y ₂ axis in the coordinate system CS1, a _2,z , b _2,z , c _2,z is the direction cosine of the Z ₂ axis of the CS2 coordinate system in the coordinate system CS1, and Coordinates for the origin O ₂ in the coordinate system CS1:

l ₁ is the distance between A ₁ and A ₂ , and l ₂ is the distance between A ₂ and O ₂ ;

Step 2: Establish a coordinate system CS3 at the lower end face of the tool holder of the multi-axis vertical milling machining center, take the center of the lower end face of the tool holder as the coordinate origin O ₃ of CS3, the tool holder axis as the Z ₃ axis of CS3, and the X ₃ axis of CS3 , Y ₃ axes are respectively parallel to X ₂ axis and Y ₂ axis of CS2 axis, the distance between the center of the lower end surface of the tool holder and the center of the lower end surface of the spindle is l ₃ +Δ ₃ , among which the design of the center of the lower end surface of the tool holder and the center of the lower end surface of the spindle The distance is l ₃ , and Δ ₃ is the manufacturing error of the tool holder; the transformation matrix M ₃₂ of CS3 and CS2 is obtained as:

Step 3: Establish coordinate system CS4, take the center of the lower end face of the collet of the multi-axis vertical milling machining center as the origin O ₄ of CS4, and the lower end face of the collet as the X ₄ Y ₄ plane Π ₁ of CS4, passing through the point O ₄ And the plane parallel to the X ₃ Z ₃ plane of the coordinate system CS3 is Π ₂ , the intersection of Π ₁ and Π ₂ is the X ₄ axis of CS4, and the collet axis is the Z ₄ axis of CS4; obtain the coordinate system CS4 to coordinate The transformation matrix of CS3 is:

Where a _4,x , b _4,x , c _4,x are the direction cosine of the X ₄ axis of the CS4 coordinate system in the coordinate system CS3, respectively, a _4,y , b _4,y , c _4,y are CS4 The direction cosine of the Y ₄ axis of the coordinate system in the coordinate system CS3, a _4,z , b _4,z , c _4,z are the direction cosine of the Z ₄ axis of the CS4 coordinate system in the coordinate system CS3; and is the coordinate of the origin O ₄ in the coordinate system CS3; considering the installation error of the collet chuck, the parameters ρ ₁ , φ ₁ , ρ ₂ , φ ₂ are used, Describe the coordinates of the origin O ₄ under CS3 for the parameters:

point Indicates the center of the upper end face of the spring chuck, l _chuck is the length of the spring chuck;

Step 4: Install the milling cutter along the axis of the lower end face of the collet, the milling cutter axis is located on the _Z4 axis of CS4, and the distance between the bottom center of the milling cutter and the CS4 origin _O4 is l _c ; establish the tool coordinate system CS5, take the milling cutter The bottom center is the CS5 origin O ₅ , and the X ₅ axis, Y ₅ axis and Z ₅ axis of CS5 are respectively parallel to the X ₄ axis, Y ₄ axis and Z ₄ axis of CS4; the transformation matrix obtained from CS5 to CS4 is:

The motion trajectory of O ₅ at the center of the bottom of the milling cutter under the coordinate system CS1 is obtained as:

and It is the coordinate of the bottom center O ₅ of the milling cutter in the coordinate system CS1;

Step 5: Install the spindle dynamic error analyzer on the multi-axis vertical milling machining center; establish the measurement coordinate system CSM, take the origin of the measurement coordinate system CSM as the measurement zero point of the spindle dynamic error analyzer, and measure the X and _M axes of the coordinate system CSM , Y _M axis and Z _M axis are respectively parallel to the X ₁ axis, Y ₁ axis and Z ₁ axis of CS1; the transformation matrix of the coordinate system CS1 and the coordinate system CSM is obtained as:

Then the coordinates of the i-th measurement point m _i measured by the spindle dynamic error analyzer in the measurement coordinate system Expressed as

and is the coordinate of the origin of CS1 in the measurement coordinate system CSM, and the measurement result of the spindle dynamic error analyzer is expressed as:

Among them, N _m is the number of measurement points of the spindle dynamic error analyzer, and z _s,1 is the installation position of the spindle dynamic error analyzer on the Z ₁ axis;

Step ₇ : Select several _{points p 1} , _{p 2 ,} . , p _n , the point with the minimum distance is selected as the corresponding point on the trajectory of the measurement point m _i ; the objective function is taken as

where m _i (x, y, z) is the coordinate of the i-th measurement point under the coordinate system CSM, and p _k (x, y, z) is the coordinate of the measurement point m _i under the coordinate system CSM of the corresponding point on the trajectory;

The objective function is optimized and solved to obtain the parameter values of the tool axis motion model, and then the motion trajectories of the bottom center _O5 of the milling _cutter and the center _O4 of the lower end face of the collet chuck are obtained according to the determined tool axis motion model _. The motion path determines the direction vector of the tool axis.

beneficial effect

The present invention solves the problem of mathematical modeling of the tool axis motion model under the consideration of the influence of the spindle motion error and the tool installation error on the axis motion of the milling tool. The established spindle error model reveals the coupling effect of the tool clamping system manufacturing and installation errors on the tool runout, establishes the quantitative and accurate relationship between each error element and the tool runout, and overcomes the disadvantage of using four parameters to describe the tool runout. Finally, after comprehensively considering the manufacturing standards of these parts, the present invention uses the method of coordinate transformation to effectively solve the problem of different parts installation error models, and clearly describes the cause of the tool runout and the movement of the spindle, in order to suppress the tool runout The theoretical basis and models are provided.

Additional aspects and advantages of the present invention will be set forth, in part, from the following description, and in part will be apparent from the following description, or may be learned by practice of the invention.

Description of drawings

The above and/or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of embodiments taken in conjunction with the accompanying drawings, wherein:

Figure 1: Schematic diagram of the definition of coordinate systems CS1 and CS2; among them: 1. Front end bearing 2. Bushing 3. Rotating spindle 4. Pull stud 5. Back end bearing;

Figure 2: Schematic diagram of three-axis vertical milling machining center and coordinate systems CS1 and CS2;

Figure 3: Schematic diagram of the definition of coordinate system CS2;

Figure 4: Schematic diagram of spindle, tool holder, collet chuck and tool;

Figure 5: The position of the collet and the schematic diagram of the definition of CS4: (a) the schematic diagram of the collet (b) the schematic diagram of the definition of CS4;

Figure 6: Schematic diagram of the definition of parameters ρ ₁ , φ ₁ , ρ ₂ , φ ₂ ;

Figure 7: Schematic diagram of the objective function;

Figure 8: Schematic diagram of objective function convergence based on particle swarm optimization;

Figure 9: Schematic diagram of tool spindle axis movement;

Figure 10: Schematic diagram of tool bottom center motion simulation and experimental results (Test No.1, Test No.2, TestNo.3, Test No.4);

Figure 11: Schematic diagram of the tool axis motion simulation results.

Detailed ways

The embodiments of the present invention are described in detail below, and the embodiments are exemplary and intended to explain the present invention, but should not be construed as a limitation of the present invention.

The machine tool used in this embodiment is Shandong Yonghua YHVT850Z, the spindle used in this machine tool is Taiwan Luoyi 1453448000 spindle, and the spindle uses a P4 precision angular contact bearing. According to the radial runout accuracy of angular contact bearings in the standard GB/T 307.2-2005, the parameter constraints 0≤a ₁ ≤4, 0≤b ₁ ≤4, 0≤a ₂ ≤4, 0≤b ₂ ≤4 can be determined , (unit: micrometer). According to the manufacturing tolerance of the spring collet in GB/T 25378-2010, the parameter constraint conditions 0≤ρ ₁ ≤20, 0≤ρ ₂ ≤20 (unit: micron) can be determined. According to GB 10944.1-2006/ISO 7388-1:1983, it can be determined that _{-300≤Δ3≤0} (unit: micron). Except for the above constraints, all other angular position constraints are [0, 2π].

The specific method steps are as follows:

Step 1: As shown in Figure 1 and Figure 2, establish a coordinate system CS1 at the installation position of the front end bearing of the three-axis vertical milling machining center, where the center of the front end bearing hole (B in Figure 1) is the coordinate origin O ₁ of CS1, The X ₁ , Y ₁ , Z ₁ axes of CS1 are respectively parallel to the X ₀ , Y ₀ , Z ₀ axes of the machine tool coordinate system CS0;

As shown in Figure 3, the spindle axis at the rear bearing position is A ₁ , and the spindle axis at the front bearing position is A ₂ . Due to the radial eccentricity error of the inner ring of the bearing, A ₁ and A ₂ will deviate from the ideal. Location. In the coordinate system CS1, the parameter equation for establishing the motion trajectory _of the spindle axis A1 at the rear bearing position is as follows:

_The parametric equation for establishing the motion trajectory of the spindle axis A2 at the position of the front end bearing is as follows

Among them, h is the distance between the installation position of the front end bearing and the rear end bearing, ω is the spindle speed; a ₁ , a ₂ , b ₁ and b ₂ are the radial parameters describing the motion of the spindle, and θ ₁ and θ ₂ are the angles describing the motion of the spindle parameter;

The coordinate system CS2 is established at the bottom end of the machine tool spindle (C in Figure 1), the center of the bottom end of the machine tool spindle is taken as the coordinate origin O ₂ of CS2, the axis direction of the spindle is the Z ₂ axis direction of CS2, and the plane P ₂ of the bottom end of the spindle is CS2 The X ₂ Y ₂ plane, the intersection of the plane P ₁ and the plane P ₂ where the two straight lines O ₁ A ₂ and A ₁ A ₂ are located is the X ₂ axis of CS2. Obviously, by calculating the outside of the Z ₂ axis of CS2 and the X ₂ axis The product can get the direction vector of the Y ₂ axis of CS2; the transformation matrix M ₂₁ from the coordinate system CS2 to the coordinate system CS1 is obtained as

where a _2,x , b _2,x , c _2,x are the cosine of the direction of the X ₂ axis of the CS2 coordinate system in the coordinate system CS1, a _2,y , b _2,y , c _2,y are the CS2 coordinate system The direction cosine of the Y ₂ axis in the coordinate system CS1, a _2,z , b _2,z , c _2,z is the direction cosine of the Z ₂ axis of the CS2 coordinate system in the coordinate system CS1, and Coordinates for the origin O ₂ in the coordinate system CS1:

l ₁ is the distance between A ₁ and A ₂ in Figure 3, and l ₂ is the distance between A ₂ and O ₂ in Figure 3; the relevant data can be obtained from the machine tool manual.

Step 2: Establish a coordinate system CS3 at the lower end face of the tool holder of the three-axis vertical milling machining center (at Figure 4D), take the center of the lower end face of the tool holder as the coordinate origin O ₃ of CS3, and the axis of the tool holder as the Z ₃ axis of CS3, The X ₃ axis and Y ₃ axis of CS3 are parallel to the X ₂ axis and Y ₂ axis of the CS2 axis respectively. The distance between the center of the lower end surface of the tool holder and the center of the lower end surface of the spindle is l ₃ +Δ ₃ , where the center of the lower end surface of the tool holder and the spindle The design distance of the center of the lower end face is l ₃ , and Δ ₃ is the manufacturing error of the tool holder; the transformation matrix M ₃₂ of CS3 and CS2 is obtained as:

Step 3: Establish the coordinate system CS4, take the center of the lower end face of the three-axis vertical milling machining center collet as the origin O ₄ of CS4, and the lower end face of the spring collet as the X ₄ Y ₄ plane Π ₁ of CS4 (Fig. 4). E), the plane passing through point O ₄ and parallel to the X ₃ Z ₃ plane of the coordinate system CS3 is Π ₂ , the intersection of Π ₁ and Π ₂ is the X ₄ axis of CS4, and the collet chuck axis is the Z ₄ of CS4 axis; the transformation matrix from coordinate system CS4 to coordinate system CS3 is obtained as:

Where a _4,x , b _4,x , c _4,x are the direction cosine of the X ₄ axis of the CS4 coordinate system in the coordinate system CS3, respectively, a _4,y , b _4,y , c _4,y are CS4 The direction cosine of the Y ₄ axis of the coordinate system in the coordinate system CS3, a _4,z , b _4,z , c _4,z are the direction cosine of the Z ₄ axis of the CS4 coordinate system in the coordinate system CS3; and is the coordinate of the origin O ₄ in the coordinate system CS3; considering the installation error of the collet chuck, the parameters ρ ₁ , φ ₁ , ρ ₂ , φ ₂ are used, Describe the coordinates of the origin O ₄ under CS3 for the parameters:

point Indicates the center of the upper end face of the collet, and l _chuck is the length of the collet.

Step 4: Install the milling cutter along the axis of the lower end face of the collet, the milling cutter axis is located on the _Z4 axis of CS4, and the distance between the bottom center of the milling cutter and the CS4 origin _O4 is l _c ; establish the tool coordinate system CS5, take the milling cutter The bottom center is the CS5 origin O ₅ , and the X ₅ axis, Y ₅ axis and Z ₅ axis of CS5 are respectively parallel to the X ₄ axis, Y ₄ axis and Z ₄ axis of CS4; the transformation matrix obtained from CS5 to CS4 is:

The motion trajectory of O ₅ at the center of the bottom of the milling cutter under the coordinate system CS1 is obtained as:

and It is the coordinate of the bottom center O ₅ of the milling cutter in the coordinate system CS1.

Step 5: Install the spindle dynamic error analyzer on the three-axis vertical milling machining center; establish the measurement coordinate system CSM, take the origin of the measurement coordinate system CSM as the measurement zero point of the spindle dynamic error analyzer, and measure the X and _M axes of the coordinate system CSM , Y _M axis and Z _M axis are respectively parallel to the X ₁ axis, Y ₁ axis and Z ₁ axis of CS1; the transformation matrix of the coordinate system CS1 and the coordinate system CSM is obtained as:

Then the coordinates of the i-th measurement point m _i measured by the spindle dynamic error analyzer in the measurement coordinate system Expressed as

and It is the coordinate of the origin of CS1 in the measurement coordinate system CSM. Considering that the spindle always rotates around the Z ₁ axis of CS1, the trajectory generated by the final tool axis motion model should also revolve around the Z ₁ axis of CS1, so use the spindle dynamic error analysis The measurement results of the instrument are expressed as:

Among them, N _m is the number of measurement points of the spindle dynamic error analyzer, and z _s,1 is the installation position of the spindle dynamic error analyzer on the Z ₁ axis.

Step ₇ : Select several _{points p 1} , _{p 2 ,} . , p _n , the point with the minimum distance is selected as the corresponding point on the trajectory of the measurement point m _i ; the objective function is taken as

where m _i (x, y, z) is the coordinate of the i-th measurement point under the coordinate system CSM, and p _k (x, y, z) is the coordinate of the measurement point m _i under the coordinate system CSM of the corresponding point on the trajectory;

The objective function is optimized and solved to obtain the parameter values of the tool axis motion model, and then the motion trajectories of the bottom center _O5 of the milling _cutter and the center _O4 of the lower end face of the collet chuck are obtained according to the determined tool axis motion model _. The motion path determines the direction vector of the tool axis.

The present invention considers that the motion error of the milling tool axis mainly comes from the manufacturing error of the spindle bearing, the tool holder and the spring chuck, so the constraint conditions of the parameters to be calibrated can be given according to the standards of different parts. In essence, the modeling of the axis motion of the milling tool requires the combined optimization of the spindle motion parameters, the tool holder installation error parameters, and the spring chuck installation error parameters that affect its movement, and finally seeks a set of optimal parameters to make the milling tool axis The motion trajectory is as close as possible to the measurement data. Here, the particle swarm optimization (PSO) is introduced into the parameter solving problem of the axis motion model of the milling tool.

The particle swarm algorithm in this embodiment successfully finds the parameter with the smallest objective function in the feasible region, so that the fitting result approximates the measurement data. In this embodiment, the number of initial particles is 500, and the dimension of the particles is 12. The velocity of any particle x _i is Each particle according to the formula

to update its speed and position in space; where t _step is the number of iterations of the current evolution; r ₁ and r ₂ are random numbers uniformly distributed between [0, 1]; c ₁ and c ₂ are acceleration constants, Usually set to 2; w is the inertia weight, vi _,d is the d-th component in the velocity vector of the ith particle, P _best,d is the d-th component in the most position vector of the ith particle, G _{best ,d} is the d-th component in the optimal position vector of the group, and x _i,d is the d-th component of the i-th particle position vector. In this paper, the algorithm terminates when the corresponding optimal value of the objective function in two consecutive iterations is less than ^10-15 . To sum up, the convergence result of the objective function obtained after applying the particle swarm optimization algorithm in the present invention is shown in Fig. 8, wherein the particle swarm algorithm takes the initial number of particles as 500, the maximum number of iterations as 1000, the maximum speed as 0.4, and the inertia weight as is 0.4.

Tool axis motion model: The motion of the tool axis can actually form a ruled surface, as shown in Figure 9. The directrix of the ruled surface is the motion trajectory of O ₅ and O ₄ , the generatrix is the straight line O ₄ O ₅ , let Then the position vector m(s,t) of any point on the tool axis motion trajectory can be expressed as the following formula:

m(s,t)=a(t)+sb(t)

Where 0≤t≤2π, 0≤s≤1.

According to the different errors of the spring chuck clamping tool, four sets of spindle dynamic error experiments are carried out in this embodiment, and the motion trajectory of the metal detection ball of the SPN 300 spindle dynamic error analyzer is used as the axis motion trajectory of the milling tool. TestNo.1, Test No.2, Test No.3 and Test No.4, the machine tool spindle structural parameters, spring chuck manufacturing error parameters and tool holder manufacturing error parameters involved in the four experiments carried out by the invention are shown in Table 1 As shown, the parameter units involved in this article are all micrometers (um).

Table 1 Structure parameters of machine tool spindle, collet chuck and tool holder

Table 2 The calibration results of the spindle motion error and the installation error parameters of the collet chuck

Although the embodiments of the present invention have been shown and described above, it should be understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and those of ordinary skill in the art will not depart from the principles and spirit of the present invention Variations, modifications, substitutions, and alterations to the above-described embodiments are possible within the scope of the present invention without departing from the scope of the present invention.

Claims (1)

1. a multi-axis milling tool axis modeling method considering tool runout is characterized in that: comprise the following steps: Step 1: Establish a coordinate system CS1 at the installation position of the front end bearing of the multi-axis milling machining center, where the center of the front end bearing hole is the coordinate origin O ₁ of CS1, and the X ₁ , Y ₁ , and Z ₁ axes of CS1 are respectively related to the machine tool coordinate system CS0. X ₀ , Y ₀ , Z ₀ axes are parallel; In the coordinate system CS1, the parameter equation for establishing the motion trajectory _of the spindle axis A1 at the rear bearing position is as follows: _The parametric equation for establishing the motion trajectory of the spindle axis A2 at the position of the front end bearing is as follows Among them, h is the distance between the installation position of the front end bearing and the rear end bearing, ω is the spindle speed; a ₁ , a ₂ , b ₁ and b ₂ are the radial parameters describing the motion of the spindle, and θ ₁ and θ ₂ are the angles describing the motion of the spindle parameter; Establish a coordinate system CS2, take the center of the bottom end of the machine tool spindle as the coordinate origin O ₂ of CS2, the axis direction of the spindle is the Z ₂ axis direction of CS2, the plane P ₂ of the bottom end of the spindle is the X ₂ Y ₂ plane of CS2, O ₁ A ₂ and the A ₁ A ₂ The intersection of the plane P ₁ and the plane P ₂ where the two straight lines are located is the X ₂ axis of CS2, and the transformation matrix M ₂₁ from the coordinate system CS2 to the coordinate system CS1 is obtained as where a _2,x , b _2,x , c _2,x are the cosine of the direction of the X ₂ axis of the CS2 coordinate system in the coordinate system CS1, a _2,y , b _2,y , c _2,y are the CS2 coordinate system The direction cosine of the Y ₂ axis in the coordinate system CS1, a _2,z , b _2,z , c _2,z is the direction cosine of the Z ₂ axis of the CS2 coordinate system in the coordinate system CS1, and Coordinates for the origin O ₂ in the coordinate system CS1: l ₁ is the distance between A ₁ and A ₂ , and l ₂ is the distance between A ₂ and O ₂ ; Step 2: Establish a coordinate system CS3 at the lower end face of the tool holder in the multi-axis milling machining center, take the center of the lower end face of the tool holder as the coordinate origin O ₃ of CS3, the tool holder axis as the Z ₃ axis of CS3, and the X ₃ axis, Y axis of CS3 The ₃ axes are respectively parallel to the X ₂ axis and the Y ₂ axis of the CS2 axis. The distance between the center of the lower end surface of the tool holder and the center of the lower end surface of the spindle is l ₃ +Δ ₃ , and the design distance between the center of the lower end surface of the tool holder and the center of the lower end surface of the spindle is l ₃ , Δ ₃ is the manufacturing error of the tool holder; the transformation matrix M ₃₂ from the coordinate system CS3 to the coordinate system CS2 is obtained as: Step 3: Establish coordinate system CS4, take the center of the lower end face of the collet of the multi-axis vertical milling machining center as the origin O ₄ of CS4, and the lower end face of the collet as the X ₄ Y ₄ plane Π ₁ of CS4, passing through the point O ₄ And the plane parallel to the X ₃ Z ₃ plane of the coordinate system CS3 is Π ₂ , the intersection of Π ₁ and Π ₂ is the X ₄ axis of CS4, and the collet axis is the Z ₄ axis of CS4; obtain the coordinate system CS4 to coordinate The transformation matrix of CS3 is: Where a _4,x , b _4,x , c _4,x are the direction cosine of the X ₄ axis of the CS4 coordinate system in the coordinate system CS3, respectively, a _4,y , b _4,y , c _4,y are CS4 The direction cosine of the Y ₄ axis of the coordinate system in the coordinate system CS3, a _4,z , b _4,z , c _4,z are the direction cosine of the Z ₄ axis of the CS4 coordinate system in the coordinate system CS3; and is the coordinate of the origin O ₄ in the coordinate system CS3; considering the installation error of the collet chuck, the parameters ρ ₁ , φ ₁ , ρ ₂ , φ ₂ are used, Describe the coordinates of the origin O ₄ under CS3 for the parameters: point Indicates the center of the upper end face of the spring chuck, l _chuck is the length of the spring chuck; Step 4: Install the milling cutter along the axis of the lower end face of the collet, the milling cutter axis is located on the _Z4 axis of CS4, and the distance between the bottom center of the milling cutter and the CS4 origin _O4 is l _c ; establish the tool coordinate system CS5, take the milling cutter The bottom center is the CS5 origin O ₅ , and the X ₅ axis, Y ₅ axis and Z ₅ axis of CS5 are respectively parallel to the X ₄ axis, Y ₄ axis and Z ₄ axis of CS4; the transformation matrix obtained from CS5 to CS4 is: The motion trajectory of O ₅ at the center of the bottom of the milling cutter under the coordinate system CS1 is obtained as: and It is the coordinate of the bottom center O ₅ of the milling cutter in the coordinate system CS1; Step 5: Install the spindle dynamic error analyzer on the multi-axis milling machining center; establish the measurement coordinate system CSM, take the origin of the measurement coordinate system CSM as the measurement zero point of the spindle dynamic error analyzer, and measure the X, _M , and Y axes of the CSM. The _M axis and the Z _M axis are parallel to the X ₁ axis, Y ₁ axis and Z ₁ axis of CS1 respectively; the transformation matrix from the coordinate system CS1 to the coordinate system CSM is obtained as: Then the coordinates of the i-th measurement point m _i measured by the spindle dynamic error analyzer in the measurement coordinate system Expressed as and is the coordinate of the origin of CS1 in the measurement coordinate system CSM, and the measurement result of the spindle dynamic error analyzer is expressed as: Among them, N _m is the number of measurement points of the spindle dynamic error analyzer, and z _s,1 is the installation position of the spindle dynamic error analyzer on the Z ₁ axis; Step ₇ : Select several _{points p 1} , _{p 2 ,} . , p _n , the point with the minimum distance is selected as the corresponding point on the trajectory of the measurement point m _i ; the objective function is taken as where m _i (x, y, z) is the coordinate of the i-th measurement point under the coordinate system CSM, and p _k (x, y, z) is the coordinate of the measurement point m _i under the coordinate system CSM of the corresponding point on the trajectory; The objective function is optimized and solved to obtain the parameter values of the tool axis motion model, and then the motion trajectories of the bottom center _O5 of the milling _cutter and the center _O4 of the lower end face of the collet chuck are obtained according to the determined tool axis motion model _. The motion path determines the direction vector of the tool axis.