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CN-121995854-A - Triaxial machine tool track contour error calculation method considering geometric and motion characteristics

CN121995854ACN 121995854 ACN121995854 ACN 121995854ACN-121995854-A

Abstract

A three-axis machine tool track contour error calculation method considering geometric and motion characteristics includes the steps of firstly establishing a three-axis machine tool geometric error equivalent mapping model, uniformly describing positioning errors, straightness errors, angle errors and inter-axis perpendicularity errors of all motion axes, mapping the geometric errors to tool nose space position errors through homogeneous coordinate transformation, secondly establishing a three-loop servo linkage control equivalent model comprising a position loop, a speed loop and a current loop, obtaining a dynamic response relation between interpolation command displacement and actual axis displacement through transfer function deduction, constructing a three-axis machine tool servo linkage control equivalent model, integrating the geometric error mapping result with three-axis linkage control dynamic position deviation to achieve prediction of tool nose actual tracks, and finally calculating the contour errors by adopting an interpolation line segment foot hanging method based on interpolation command position points.

Inventors

  • ZHANG XING
  • Chang Tongyu
  • MEI CHAOJIE
  • Cao Fangcun
  • ZHAO WANHUA

Assignees

  • 西安交通大学

Dates

Publication Date
20260508
Application Date
20260210

Claims (5)

  1. 1. The method for calculating the track contour error of the triaxial machine tool by considering geometric and motion characteristics is characterized by comprising the following steps of: The method comprises the steps of 1) establishing a three-axis machine tool geometric error equivalent mapping model, namely establishing the three-axis numerical control machine tool geometric error model based on a multi-rigid body system theory, and carrying out unified matrix description on positioning errors, straightness errors, angle errors and inter-axis perpendicularity errors of all motion axes; Step 2) establishing a triaxial machine tool servo linkage control equivalent model, namely establishing a tricyclic control equivalent model comprising a position ring, a speed ring and a current ring aiming at a uniaxial servo system of the numerical control machine tool, and modeling transfer functions of a driver, a servo motor and a mechanical transmission link; Step 3) the integrated model of the geometric errors and the linkage control errors and the actual track point prediction are carried out, namely the tool nose geometric error mapping result obtained in the step 1) and the three-axis linkage control dynamic position deviation obtained in the step 2) are integrated uniformly, and under the same interpolation time sequence and coordinate system, superposition calculation is carried out on each axis error, so that the actual space position prediction value of the tool nose at each interpolation instruction position is obtained, and the integrated modeling of the geometric errors and the linkage control errors and the actual track point prediction are realized; And 4) finally, taking the expected track formed by interpolation instruction position points as a reference profile, taking the actual position points of the tool tip points predicted by the integrated model as deviation points, calculating the shortest distance between the actual position points and the expected track by an interpolation line segment foot drop method, and taking the distance as the profile error at the corresponding position, thereby realizing the rapid calculation of the profile error without track analysis and offline measurement.
  2. 2. The method for calculating the trajectory profile error of the three-axis machine tool taking geometric and motion characteristics into consideration according to claim 1, wherein the specific process of the step 1) is as follows: 1.1 A transformation matrix of the tool coordinate system to the workpiece coordinate system under ideal conditions: Classifying the three-axis machine tool into FXYZ, XFYZ, XYFZ and XYZF according to the sequence from the workpiece to the tool, wherein the letter F represents a fixed part of the machine tool, namely a machine tool coordinate system; Establishing a three-axis machine tool geometric error model by adopting XYFZ, wherein in an initial state, a machine tool body coordinate system is F, a local coordinate system X, Y, Z, S, T, W is respectively established based on the F direction, wherein a main axis coordinate system is S, a tool coordinate system is T, and W represents a workpiece coordinate system; Under ideal conditions, T represents a coordinate transformation matrix, the upper and lower marks on the left side of the matrix represent the transformed original coordinate system subscript and the target coordinate system subscript, the upper right corner mark i represents the transformation matrix in an ideal state, and assuming that the workbench coordinate system X, Y, Z sequentially moves along the respective axial directions by distances X, Y and Z, for a XYFZ type machine tool, the motion chain from the workpiece coordinate system to the machine tool coordinate system is W-X-Y-F-Z-S-T, and the motion chain is represented by the following formula: Wherein: is a transformation matrix of the object coordinate system W relative to the X coordinate system, Is a transformation matrix of an X coordinate system relative to a Y coordinate system, Is a transformation matrix of the Y coordinate system relative to the lathe bed coordinate system F, Is a transformation matrix of the bed coordinate system F relative to the Z coordinate system, Is a transformation matrix of the Z coordinate system relative to the main axis coordinate system S, In an ideal state, the principal axis S and the Z axis, the tool T and the principal axis S, the workbench W and the X axis do not have relative motion, namely 、 、 The motion quantity is a unit matrix, and x, y and z are nominal motion quantities of a translation axis, namely instruction motion quantities sent by a numerical control system; 1.2 Consider the transformation matrix of the tool coordinate system to the workpiece coordinate system under geometric error conditions: each axis of the machine tool has 6 geometric errors, a perpendicularity error exists between the two axes, and 21 geometric errors are taken into consideration, and a transformation matrix from a tool coordinate system to a workpiece coordinate system under the action of the geometric errors established in motion among bodies is shown as follows: wherein the upper right corner r represents the transformation matrix taking into account the geometrical error, To consider the transformation matrix of the object coordinate system W with respect to the X coordinate system in the geometric error state, To consider the transformation matrix of the X-coordinate system with respect to the Y-coordinate system in the geometric error state, To consider the transformation matrix of the Y-coordinate system with respect to the bed coordinate system F in the geometric error state, In order to consider the transformation matrix of the bed coordinate system F with respect to the Z coordinate system in the geometrical error state, To consider the transformation matrix of the Z-coordinate system with respect to the principal axis coordinate system S in the geometric error state, In order to consider the transformation matrix of the spindle coordinate system S relative to the workpiece coordinate system T under the geometric error state, the spindle S and the Z axis, the tool T and the spindle S and the workbench W and the X axis have no relative motion under the ideal state, namely 、 、 Is a unit matrix; ideal conversion matrix And a transformation matrix taking into account geometrical errors The relationship between them is an ideal conversion matrix Right-hand-by-one error motion matrix The following formula is shown: Based on small error assumption, error motion change matrix from tool coordinate system T to workpiece coordinate system W Assume the following formula: Wherein: 、 、 for the spatial position error of the actual cutting point of the tool relative to the ideal cutting point, 、 、 The rotation error of the actual cutting point of the cutter relative to the ideal cutting point is obtained; the spatial position error portion of the error matrix is shown as follows: the amount of Δx, Δy, and Δz error motion is a position-dependent quantity; 1.3 Unified space error model of three-axis milling machine with four structures: The kinematic chain of the 4 three-axis machine tool structure is as follows: FXYZ:W→F→X→Y→Z→S→T; XFYZ:W→X→F→Y→Z→S→T; XYFZ:W→X→Y→F→Z→S→T; XYZF:W→X→Y→Z→F→S→T. The homogeneous coordinates on the right side of the machine tool coordinate system F are converted into positive transformation, and the transformation on the left side is negative transformation, so that a singular function shown in a formula (6) is introduced, and the singular function can automatically judge whether the type of the transformation matrix is positive transformation or negative transformation according to the corresponding machine tool type; Wherein: when (when) Time of day Parameter i=0, 1,2,3, representing 4 types of machine tools FXYZ, XFYZ, XYFZ and XYZF, respectively; conversion matrices of X, Y and Z axes respectively corresponding to =0, 1,2, i.e Corresponding to =0, Corresponding to =1, Corresponding to =2; By using the singular matrix of the matrix, Restated, they are conversion matrices , And Is a unified expression form of (a); will unify the transformation matrix , And And carrying out the method into the formula (2) to obtain a unified space error model of 4 triaxial machine tools, wherein the formula (7) is a displacement error part of the unified kinematic model: Wherein: For the positioning error of the X-axis, Is the error of the horizontal straightness of the X axis, Is the error of the vertical straightness of the X axis, Is the pitch angle error of the X-axis, Is the deflection angle error of the X-axis, Is the rolling angle error of the X axis; For the positioning error of the Y-axis, Is the error of the horizontal straightness of the Y axis, Is the error of the vertical straightness of the Y axis, Is the pitch angle error of the Y-axis, Is the deflection angle error of the Y axis, The rolling angle error is Y-axis; For the positioning error of the Z axis, Is the error of the horizontal straightness of the Z axis, Is the error of the vertical straightness of the Z axis, Is the pitch angle error of the Z axis, Is the error of the deflection angle of the Z axis, The Z-axis rolling angle error; , , the vertical error between the X axis and the Y axis, the X axis and the Z axis respectively.
  3. 3. The method for calculating the trajectory profile error of the three-axis machine tool taking geometric and motion characteristics into consideration according to claim 2, wherein the specific process of the step 2) is as follows: 2.1 Equivalent transfer function of each link of the servo system: 1) Equivalent transfer function of permanent magnet synchronous motor: Selecting Control=0 as core control architecture of PMSM servo system, knowing voltage equation of motor under dq0 coordinate by mathematical model of permanent magnet synchronous motor, adopting vector control strategy When=0 control mode, the voltage equation reduces to: Wherein: For the d-axis voltage to be the same, For the q-axis voltage to be applied, For the q-axis inductance, For the q-axis current of the motor, The armature resistance of the motor; the transfer function of q-axis armature current and q-axis voltage obtained by Laplace transform of equation (8) is: By using When the control strategy is=0, the direct-axis component of the stator current is completely suppressed, which is equivalent to the open-circuit state of the direct-axis winding, and for the surface-mounted permanent magnet synchronous motor, the control mode motor port characteristic is equivalent to that of a separately excited DC motor, and the motor output torque and the quadrature-axis current In a linear relationship, namely: I.e. q-axis current and electromagnetic torque The transfer function of (2) is: Wherein: Is the electromagnetic moment of the motor, For the pole pair number of the motor, Is the flux linkage amplitude of the permanent magnet; 2) Equivalent transfer function of servo driver: In the current loop control framework, a controlled object is formed by a PWM inverter and a PMSM armature loop, wherein the PWM inverter is equivalently modeled as a first-order inertia link with pure hysteresis characteristic, and the dynamic characteristic of the PWM inverter is formed by a time constant And equivalent gain coefficient The phase delay of the link is derived from three delay factors, namely PWM carrier comparison time, power device switch dead time and inverter power device switch turn-off delay time, wherein the accumulated limit value of the delay components does not exceed a single PWM modulation period, and the transfer function of the delay link is expressed as Combining inverter gains The transfer function of a PWM inverter is expressed as: Wherein: In the case of a q-axis voltage command, (S) is a PWM modulation period, For the operating frequency (Hz) of the inverter switching tubes, For inverter gain, approximately 1; For a pair of Performing a first-order Taylor expansion approximation on Pair at=0 Taylor expansion due to And (3) the value of (3) is very small, gao Jiexiang is ignored, a first-order term is reserved, and finally, the transfer function of the PWM inverter is approximated to a first-order inertia link: 3) Transfer function of mechanical transmission system: The dynamic equation is obtained according to the equivalent dynamic model of the two-inertia two-degree-of-freedom ball screw mechanical system: Order the , Then transform equation 14) to: the Laplace transform is carried out on the above formula to obtain: The angular speed of the motor with two inertia degrees and two degrees of freedom is obtained through the derivation With respect to motor electromagnetic torque The transfer function of (2) is: Similarly, the angular velocity of the screw is obtained With respect to motor electromagnetic torque The transfer function of (2) is: Wherein: For the moment of inertia of the rotor of the motor, Is the equivalent rotational inertia of the screw rod, In order to act on the torque of the motor rotor, In order to act on the screw torque, For the rotational displacement of the rotor of the motor, For the screw rod to be in torsion displacement, For the angular velocity of the screw rod, For the angular velocity of the motor, For the torsional rigidity of the coupling, For the purpose of torsional damping of the rotor of the electric machine, For the purpose of torsional damping of the coupling, For the purpose of bearing torsional damping, , , , ; 2.2 Single-axis three-ring control model establishment: The servo system adopts three-loop control, namely a current loop, a speed loop and a position loop, wherein the current loop and the speed loop use PI controllers, the position loop uses P controllers, and a closed-loop control structure of the single-shaft servo system is established according to the obtained equivalent transfer functions of all links, wherein the open/closed-loop transfer functions of the three control links are expressed as: 1) Current loop transfer function: According to the control structure, the open loop transfer function of the current loop is obtained as follows: the current closed loop transfer function is: Wherein: , , , , , , is the current loop controller proportional gain (V/a), Integrating a time constant for the current loop controller; 2) Speed loop transfer function: The speed loop open-loop transfer function is deduced from a current closed-loop transfer function formula (20) and a transfer function formula (17) of motor angular speed and motor electromagnetic torque, and is: and further deriving a speed closed loop transfer function as: Wherein: In order to be a speed command, Is the torque constant of the motor and is equal to the torque constant of the motor, A/(rad/s) is the proportional gain (A/(rad/s)) of the speed loop controller, The time constant is integrated for the speed loop controller, , , , , , , , , , , , ; 3) Position loop transfer function: From the speed loop closed loop transfer function equation (22), and the position controller transfer function, the position loop open loop transfer function is calculated as: and further obtaining a closed loop transfer function of the position loop: Wherein: as a result of the actual position of the device, In order to be a position of the command, Proportional gain (Hz) for the position loop controller, , , , , , , , , ; 4) Transfer function of disturbance input: for a single axis servo system, the input is assumed to be a disturbance , According to the closed loop transfer function of the current And a transfer function of motor angular velocity and motor electromagnetic torque Obtaining disturbance Input and rotational speed output of motor Is a transfer function of (2): Wherein: , , , , , , , , , , , , , , , , , , , ; Disturbance of And a stage displacement output Is a transfer function of (2): Wherein: , , , , , , , , , , , 。
  4. 4. A method for calculating a three-axis machine tool trajectory profile error taking geometric and motion characteristics into consideration as defined in claim 3, wherein the specific process of step 3) is as follows: 3.1 Multiaxial linkage feed motion error resolution: The mechanical and electrical integrated model of the feeding system is constructed by analyzing a physical mechanism of servo driving of a single-shaft feeding system, and further integrating a motion control system, a servo amplifier, a PWM rectifying inverter, a three-phase permanent magnet synchronous motor and a mechanical transmission link, wherein an X-axis typical ball screw mechanical transmission system is adopted, and the electromagnetic torque output by the motor is finally obtained by deducting the magnetomotive force of each phase winding and the three-phase synthetic magnetomotive force of the stator and the rotor through establishing a voltage and current equation of the stator and the rotor for the three-phase permanent magnet synchronous motor, and the following formula is adopted: Wherein: Is a nominal moment, is generated after the numerical control interpolation instruction is input, Harmonic moment generated for non-ideal and electrical nonlinearity of the electromechanical structure, Is the magnetic permeability of the air, and the air is the air, Indicating the diameter of the rotor, For the axial length of the rotor core, Is the number of the magnetic pole pairs of the motor rotor, For the length of the air gap, The magnetomotive force is synthesized for the stator, The magnetomotive force is synthesized for the rotor, Is the included angle between the magnetomotive force axes of the stator and the rotor; further, the position loop feedback and velocity loop feedback disturbances re-enter the closed loop control loop, eventually creating redundant displacement responses at the machine end tool and table, namely: furthermore, under the combined action of the numerical control instruction, feedback interference of a position loop and a speed loop, motor torque harmonic waves and other excitation sources, the final displacement output of the workbench is expressed as follows: Wherein: for the displacement of the table without load disturbance, The position of the working table is instructed, As a function of the closed loop transfer of the position, For the harmonic moment disturbance-position output transfer function, For the displacement of the table without load disturbance, Table displacement fluctuations for motor harmonic torque excitation, Feedback interfering with the displacement response generated by the closed loop control circuit at the end of the machine for the position loop and the velocity loop; For other linear feed shafts of the triaxial machine tool, the method is adopted to obtain the motion displacement response of the mechanical execution end along the feed direction of each shaft, and the actual displacement output is subtracted from the ideal displacement to obtain the transient state and steady state following errors during the feed of each shaft, wherein the following formula is shown as follows: Wherein: as a follow-up error in transient and steady states, For the actual displacement output, Outputting the displacement for the instruction; 3.2 Holographic state solution of geometry-linkage process: Establishing an integrated model of the geometric error and the linkage control error of the numerical control machine tool, and realizing dynamic displacement response solution of the tool tip under the influence of multi-link interactive coupling, wherein the solution is shown in the following formula; Wherein: to account for geometrical errors and linkage errors in the tip displacement bias, As the deviation of the nose point caused by the geometrical error, To the deflection of the nose point caused by the linkage error, As the following error of the X-axis, As the following error of the Y-axis, As the following error of the Z-axis, Is a conversion matrix of geometric errors between a global coordinate system and a dynamic coordinate system of a workpiece, The transformation matrix is a transformation matrix of the multi-axis linkage error between the global coordinate system and the dynamic coordinate system of the workpiece; And solving the integrated model by adopting an off-line time domain discrete numerical calculation method, inputting an interpolation instruction point of the processing track, and outputting a predicted tool tip point taking geometrical errors and linkage errors into consideration.
  5. 5. The method for calculating the trajectory profile error of the three-axis machine tool taking geometric and motion characteristics into consideration according to claim 4, wherein the specific process of the step 4) is as follows: solving the trace outline error by interpolation line segment foot drop method, namely setting the current actual position point The corresponding instruction position point is First find the current interpolation position The previous n interpolation points are stored to obtain n-1 interpolation line segments, and then the actual position points are passed Making the perpendicular lines of the n-1 interpolation line segments and finding the drop foot (I=1, 2,3.., n-1), if the drop foot is within the interpolation line segment, storing a distance from the actual point to the drop foot point, and if the drop foot is outside the interpolation line segment, taking a distance from the actual point to a start point of a vector of the corresponding interpolation line segment of the drop foot as a distance from the interpolation line segment to the actual position point; the calculation of the drop foot point uses a vector projection method: (1) Obtaining n-1 interpolation vectors for n interpolation points , , , … , Additionally obtain the vector from n-1 instruction points to actual points , , , … , ; (2) Calculating a projection coefficient t, namely the projection ratio of the vector from the instruction point to the actual point on each interpolation vector: Limiting the range of projection coefficients t: When the drop foot point is not in the interpolation line segment, taking the starting point of the interpolation vector as the closest point; (3) Obtaining projection coefficients to obtain interpolation line segments Corresponding drop foot : Wherein: for the current instruction location point Instruction location points of the first n interpolation cycles, For interpolating line segments Corresponding drop feet; Finally, obtain Distance to the n-1 interpolation line segments (I=1, 2,3.., n-1), wherein the minimum distance I.e. the actual point Profile error values of (a), namely: Wherein: for the current actual position point A corresponding value of the contour error, As the point of the current actual position, For the current actual position point to the foot of the n interpolation line segments, The current actual position point is modulo the vector of the foot of the n interpolation line segments.

Description

Triaxial machine tool track contour error calculation method considering geometric and motion characteristics Technical Field The invention relates to the technical field of precision detection and simulation of numerical control machine tools, in particular to a method for calculating a triaxial machine tool track contour error by considering geometric and motion characteristics. Background In modern numerical control machine tool machining, geometric errors and linkage errors of a control system are core factors influencing machining precision. The geometric errors comprise straightness errors, angle errors and the like, and the ISO 230-1 standard divides the geometric errors into position-related geometric errors and position-independent geometric errors, so that the geometric errors are key to machine tool precision analysis and compensation. The current geometric error modeling is based on static error data, and dynamic behaviors of actual machining of a machine tool are difficult to consider through analysis of a relative motion model of a multi-rigid-body system, and linkage errors cannot be accurately described. The servo system of the machine tool usually adopts a three-loop control structure of a position loop, a speed loop and a current loop, but the existing control model models multiple stress and single axis, and ignores the error coupling effect in multi-axis linkage and the influence of geometric errors on the actual motion trail. The geometrical error or linkage error is considered independently, so that the actual track of the tool nose point cannot be calculated accurately, and the contour error of the part is difficult to predict accurately. Along with the development of numerical control technology, error prediction is turned from hardware compensation to numerical simulation, but the existing method （Fan K G, Yang J G, Yang L Y. Unified Multi-Error Model for Four Types CNC Machining Center[C]. Applied Mechanics and Materials. 2013, 395: 1162-1169） based on finite element analysis and kinematic modeling is complex in calculation, poor in instantaneity, and difficult to widely apply in actual production due to single error modeling. Meanwhile, most of the existing contour error predictions lack real-time prediction capability in the processing process by offline analysis （Sheng X J, Wang L F. A Comparison Strategy for Improving the Precision of Contour Error Estimation[J]. International Journal of Precision Engineering and Manufacturing, 2019, 20(8): 1395-1403）,. In summary, the prior art lacks an effective integrated model, cannot comprehensively consider geometric errors and linkage control errors, and is difficult to quickly calculate the contour errors of each point in the processing process based on numerical control instructions under the condition of not relying on online measurement. Disclosure of Invention In order to overcome the defects in the prior art, the invention aims to provide a three-axis machine tool track contour error calculation method considering geometric and motion characteristics, which is used for rapidly predicting the contour error of a machining track based on a numerical control system interpolation instruction point under the condition of not depending on online measurement by integrating a three-axis machine tool geometric error model and a single-axis servo linkage control model. In order to achieve the above purpose, the invention adopts the following technical scheme: a method for calculating the track contour error of a triaxial machine tool by considering geometric and motion characteristics comprises the following steps: The method comprises the steps of 1) establishing a three-axis machine tool geometric error equivalent mapping model, namely establishing the three-axis numerical control machine tool geometric error model based on a multi-rigid body system theory, and carrying out unified matrix description on positioning errors, straightness errors, angle errors and inter-axis perpendicularity errors of all motion axes; Step 2) establishing a triaxial machine tool servo linkage control equivalent model, namely establishing a tricyclic control equivalent model comprising a position ring, a speed ring and a current ring aiming at a uniaxial servo system of the numerical control machine tool, and modeling transfer functions of a driver, a servo motor and a mechanical transmission link; Step 3) the integrated model of the geometric errors and the linkage control errors and the actual track point prediction are carried out, namely the tool nose geometric error mapping result obtained in the step 1) and the three-axis linkage control dynamic position deviation obtained in the step 2) are integrated uniformly, and under the same interpolation time sequence and coordinate system, superposition calculation is carried out on each axis error, so that the actual space position prediction value of the tool nose at each interpolation instruction position is ob