Feedforward compensation and adaptive fuzzy PID (Proportion Integration Differentiation) integrated hybrid control method

Abstract

The invention discloses a feed-forward compensation and adaptive fuzzy PID integrated hybrid control method, and relates to the technical field of automatic control. The feed-forward compensation and self-adaptive fuzzy PID integrated hybrid control method comprises the following steps: real-time data acquisition: acquiring position errors and change rates of all shafts; feed-forward compensation calculation: a feed-forward compensation module is used for generating a feed-forward compensation amount through inverse kinematics; fuzzy reasoning: a self-adaptive fuzzy PID module is used for calculating PID parameter increment; predictive compensation: establishing a multi-axis linkage predictive model, and generating prospective predictive compensation; and controlled quantity synthesis. According to the hybrid control method fusing feedforward compensation and adaptive fuzzy PID, the following error data of each driving shaft is collected in real time, a contour error source is reversely deduced in combination with a kinematic model, a multi-shaft linkage prediction model is constructed, and accurate prediction of the tool nose point motion error in a multi-shaft linkage state is realized through Matlab / Simulink simulation.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, specifically to a hybrid control method that integrates feedforward compensation and adaptive fuzzy PID. Background Technology

[0002] Currently, high-performance five-axis linkage high-end CNC machine tools, especially precision machine tools for machining complex curved surface parts in the aerospace field, typically consist of three linear axes (X / Y / Z) and two rotary axes (A / C or B / C). They are equipped with a high-rigidity gantry structure, linear motors or dual-drive synchronous technology, a fully closed-loop grating ruler feedback system, and an electric spindle technology to achieve high speeds of over 20,000 rpm. The CNC system adopts a multi-channel, multi-axis linkage control architecture and is equipped with RTCP (Rotary Tool Center Point) function to achieve precise control of the tool tip trajectory.

[0003] However, current high-performance five-axis linkage high-end CNC machine tools still have the following requirements for core technologies: The complex kinematic transformation of a five-axis linkage machine tool amplifies the impact of the following error of each axis on the contour accuracy, and it is necessary to accurately decompose the contour error to each axis through feedforward compensation; When machining thin-walled parts such as impellers and casings, adaptive fuzzy PID can effectively cope with elastic deformation caused by changes in cutting force and suppress chatter through parameter self-tuning. The backlash and nonlinear friction of the rotating shaft will produce periodic contour errors during spatial interpolation, which need to be compensated for in advance by the predictive model.

[0004] To address the aforementioned technical requirements, this invention proposes a hybrid control algorithm that integrates feedforward compensation and adaptive fuzzy PID. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a hybrid control method that integrates feedforward compensation and adaptive fuzzy PID, thus resolving the defects and deficiencies in existing technologies.

[0006] To achieve the above objectives, the present invention provides the following technical solution: A hybrid control method integrating feedforward compensation and adaptive fuzzy PID includes the following steps: Step 1, Real-time Data Acquisition: Obtain the position error of each axis. and rate of change ; Step 2, Feedforward Compensation Calculation: This includes a feedforward compensation module that generates the feedforward compensation amount through inverse kinematics. ; Step 3, Fuzzy Inference: This includes an adaptive fuzzy PID module that calculates the increments of PID parameters. , , ; Step 4, Prediction Compensation: Establish a multi-axis linkage prediction model, and generate forward prediction compensation based on steps 1, 2, and 3. u comp ; Step 5, control quantity synthesis:

[0007] in: For adaptive fuzzy PID output, This is the feedforward compensation amount. For forward forecasting compensation; Step 6, Closed-loop execution: Output control input and update system status.

[0008] Preferably, the feedforward compensation module is based on multi-axis kinematics inverse analysis to construct an error mapping model, including error inverse derivation and nonlinear compensation model; Error back-end derivation: contour error Decomposed into individual axes: ; Let be the Jacobian matrix, describing the differential relationship between the tool tip error and the errors of each axis: ; in: Generalized coordinate vector It includes linear axes and rotational axes. x, y, z The coordinate vectors of the linear axes. a, c The coordinate vectors of the rotation axis; The tool tip error vector ; Nonlinear compensation model: ; In the formula, Gain compensation for reverse gap; Coulomb friction coefficient; It is a friction speed sensitive factor; This is an estimate of the inter-shaft elastic deformation. It is a hyperbolic tangent function with an output range of (-1, 1) and a smooth S-shaped curve.

[0009] Preferably, the adaptive fuzzy PID module is used for real-time acquisition of position error. and rate of change It can be adjusted online through 49 fuzzy rules, including fuzzification processing, fuzzy rule base, and parameter updates; Blur processing: Input normalization: , ; Trigonometric membership functions: In the formula, Center point (NB=-1, PB=1). ; Fuzzy rule base: Typical rule example: IF is NB AND is NB THEN is PB, is NB, is PM; Weighted average solution fuzzy: , ; Parameter update: The update expression is represented as: ; In the formula, the Clip function (also known as the clipping function) is mathematically defined as follows: ; Preferably, establishing a multi-axis linkage prediction model includes the construction of a state-space prediction model and look-ahead compensation; Construction of the state-space prediction model: The core components are: ; In the formula, , For the system matrix; Inter-axis coupling weights; This is a nonlinear disturbance term; The linear dynamic term describes the error of the system at the previous time step. Impact on the next moment; As a control input, it is used to quantify changes in the control quantity. The ability to correct errors; This refers to inter-axis coupling terms. It is a nonlinear function; q represents the generalized coordinate, a vector describing the position of each axis of the machine tool, including linear axes. X, Y, Z Displacement and rotation axes a, c or b The rotation angle of the axis is expressed as q is a generalized velocity, representing the vector of velocities along each axis; Forward compensation: ; In the formula, To compensate for the gain matrix, a 3-step look-ahead is set.

[0010] This invention provides a hybrid control method that integrates feedforward compensation and adaptive fuzzy PID, which has the following advantages: 1. This application constructs a multi-axis linkage prediction model by real-time acquisition of the following error data of each drive axis and reverse deduction of the contour error source by combining the kinematic model. Through Matlab / Simulink simulation, the error between the result and the actual machine tool test data does not exceed 5%, thus realizing accurate prediction of the tool tip motion error in multi-axis linkage state.

[0011] 2. This application achieves high-precision motion control through the synergistic effect of feedforward compensation and adaptive fuzzy PID. The feedforward compensation module, based on a multi-axis kinematic model, dynamically generates compensation amounts to offset the nonlinear errors of the mechanical transmission chain, including backlash, elastic deformation, and frictional disturbances, by reverse-analyzing the sources of following errors for each drive axis. An error mapping model is established to decompose the contour error to each axis for targeted compensation. The adaptive fuzzy PID module dynamically adjusts the proportional, integral, and derivative parameters using a fuzzy inference mechanism based on real-time acquired position error data. K p , K i , K d The fuzzy rule base optimizes the PID gain online based on the error magnitude and rate of change, realizing intelligent adjustment of control parameters, ensuring fast response and suppressing overshoot. The multi-axis linkage prediction model integrates kinematic and real-time error data to construct a prediction algorithm for the trajectory deviation of the blade tip, and corrects the command trajectory in advance through look-ahead control. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of the IAE and ISE values ​​of the hybrid control method that integrates feedforward compensation and adaptive fuzzy PID described in this invention. Detailed Implementation

[0013] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0014] Example: This invention provides a hybrid control method that integrates feedforward compensation and adaptive fuzzy PID. This hybrid control algorithm eliminates systematic errors through feedforward compensation, adapts to dynamic disturbances through fuzzy PID, and solves multi-axis coupling problems through a predictive model. Specifically, it includes the following steps: Step 1, Real-time Data Acquisition: Obtain the position error of each axis. and rate of change ; Step 2, Feedforward Compensation Calculation: This includes a feedforward compensation module that generates the feedforward compensation amount through inverse kinematics. ; Step 3, Fuzzy Inference: This includes an adaptive fuzzy PID module that calculates the increments of PID parameters. , , ; Step 4, Prediction Compensation: Establish a multi-axis linkage prediction model, and generate forward prediction compensation based on steps 1, 2, and 3. u comp ; Step 5, control quantity synthesis:

[0015] in: For adaptive fuzzy PID output, This is the feedforward compensation amount. For forward forecasting compensation; Step 6, Closed-loop execution: Output control input and update system status.

[0016] In this embodiment, the feedforward compensation module constructs an error mapping model based on multi-axis kinematics inverse analysis, including error inverse derivation and nonlinear compensation model; Error back-end derivation: contour error Decomposed into individual axes: ; The Jacobian matrix is ​​the core of the machine tool kinematics differential mapping, describing the differential relationship between the tool tip error and the errors of each axis: ; in: Generalized coordinate vector It includes linear axes and rotational axes. x, y, z The coordinate vectors of the linear axes. a, c The coordinate vectors of the rotation axis; The tool tip error vector ; Table 1: The physical meaning of each matrix element is as follows:

[0017] Nonlinear compensation model: ; In the formula, Gain compensation for reverse gap; Coulomb friction coefficient; It is a friction speed sensitive factor; This is an estimate of the inter-shaft elastic deformation. It is a hyperbolic tangent function with an output range of (-1, 1) and a smooth S-shaped curve.

[0018] In this embodiment, the adaptive fuzzy PID module is used to collect position errors in real time. and rate of change It can be adjusted online through 49 fuzzy rules, including fuzzification processing, fuzzy rule base, and parameter updates; Blur processing: Input normalization: , ; Trigonometric membership functions: In the formula, Center point (NB=-1, PB=1). ; Fuzzy rule base: Typical rule example: IF is NB AND is NB THEN is PB, is NB, is PM; Weighted average solution fuzzy: , ; Parameter update: The update expression is represented as: ; In the formula, the Clip function (also known as the clipping function) is mathematically defined as follows: ; In adaptive fuzzy PID control, its role is to ensure the safety and stability of parameter updates through hard constraint boundaries.

[0019] The implementation process of adaptive fuzzy PID control is as follows: 1) Dynamic adjustment mechanism The control state, including errors, is evaluated in real time through a fuzzy inference system. and error change rate The incremental adjustment of the PID parameters is generated. , , ).

[0020] Specifically: When the error is large, the response is enhanced, then when For negative large (NB) and When the value is negative (NB), the proportional gain is significantly increased. To quickly reduce errors; when the error is small, oscillations are suppressed, then when When approaching zero (ZE), reduce the integral gain ( Avoid overshooting; 2) Weighted average solution to fuzzy The output of all activation rules: Contribution of each rule Take the minimum value of the input membership degree. The final increment value is the central value of each rule's consequent. Weighted average, for example (PB) represents the maximum positive adjustment; 3) Constrained update strategy The momentum update method combined with boundary constraints is used:

[0021] The momentum term can have different values, such as By retaining 40% of historical values ​​and 60% of new values, boundary constraints are applied to smooth parameter fluctuations to prevent parameters from going out of bounds and causing system instability. In the above formula, according to the boundary protection mechanism, For example: when At that time, forced ; when At that time, forced ; like ,current After the update ; Furthermore, in this embodiment, establishing a multi-axis linkage prediction model includes the construction of a state-space prediction model and look-ahead compensation; Construction of the state-space prediction model: The core components of the state-space prediction model are: ; In the formula, , For the system matrix; Inter-axis coupling weights; This is a nonlinear disturbance term; The linear dynamic term describes the error of the system at the previous time step. Impact on the next moment; Specifically, it is the state transition matrix, which reflects the inherent dynamic characteristics of the machine tool, such as the inertia of the mass, springs, and damping system, and the system matrix. The system matrix can be obtained by exciting each axis with white noise, acquiring the frequency response function (FRF), and then using the subspace identification method. This can be obtained through system identification experiments; As a control input, it is used to quantify changes in the control quantity. The ability to correct errors; For inter-axis coupling terms, the coupling weight matrix is... The circular / helical trajectory can be executed through multi-axis synchronous motion testing and obtained by least squares fitting method. It is a nonlinear function, containing coupling effects such as Coriolis force and centrifugal force, as well as higher-order disturbance terms, such as those involving cutting force disturbances under time-varying loads, nonlinearity of guideway friction, or return errors of harmonic reducers; q represents a generalized coordinate, a vector describing the position of each axis of the machine tool. For a five-axis machine tool, this includes linear axes. X, Y, Z Displacement and rotation axes a, c or b The rotation angle of the axis is expressed as q is a generalized velocity, representing the vector of velocities along each axis; Forward compensation: ; In the formula, To compensate for the gain matrix, a 3-step look-ahead is set.

[0022] Table 2: Physical meaning of each parameter in the implementation steps of this embodiment:

[0023] In this embodiment: The feedforward compensation module is based on inverse analysis of multi-axis kinematics to construct an error mapping model to offset nonlinear errors such as backlash and elastic deformation. The key parameters are the transmission chain stiffness coefficient and friction model parameters. Error inverse derivation decomposition is used to project the contour error vector onto the coordinate system of each drive axis to realize the dynamic distribution of error between axes. The key parameters are the inter-axis coupling coefficient and the Jacobian matrix. The adaptive fuzzy PID module is used to acquire position errors in real time. e and rate of change de It can be dynamically adjusted online using 49 fuzzy rules. K p ,K i , K d Parameters, the key parameters are the membership function range and the rule weights; Multi-axis predictive modeling is used to fuse kinematic equations with historical error data to construct an ARMA predictive model, predicting trajectory deviations 1-3 interpolation cycles in advance. The key parameters are the prediction time domain and the error decay factor. A multi-axis linkage prediction model is established to correct the G-code instruction points in reverse based on the prediction results, thereby reducing the phase lag of the contour error. The key parameters are look-ahead step size and compensation gain. The control quantity synthesis is used to weight the feedforward compensation quantity and the adaptive fuzzy PID output quantity to achieve the synergistic effect of static error compensation and dynamic error suppression. The key parameter is the weighting factor, which can be set to 0.3 to 0.7 in this embodiment.

[0024] Furthermore, in this embodiment, programming is performed using MATLAB: for the adaptive fuzzy PID module, the proportional, integral, and derivative parameters are dynamically adjusted using fuzzy inference mechanisms based on real-time collected position error data. K p , K i , K d The fuzzy rule base optimizes the PID gain online based on the error magnitude and rate of change, enabling intelligent adjustment of control parameters, ensuring both fast response and suppressing overshoot.

[0025] The specific implementation process is as follows: An adaptive PID control system based on Mamdani-type fuzzy inference was constructed using MATLAB M-code, primarily for real-time adjustment of PID parameters. K p , K i , K d To improve the accuracy of dynamic control.

[0026] The system uses error ( e ) and error change rate ( de The input variable is fuzzified using triangular membership functions of seven fuzzy subsets (NB to PB), and the output is the incremental adjustment of the PID parameters. , , The fuzzy partitioning of input and output variables covers the entire range from negative to positive, with inputs normalized to [-1, 1] and outputs limited to the intervals [-0.3, 0.3], [-0.2, 0.2], and [-0.1, 0.1] respectively. The rule base contains 49 fuzzy rules, which correlate input and output through "IF-THEN" logic. For example, when the error is negative (NB) and the rate of change of the error is negative (NB), the proportional gain is significantly increased (…). =PB) and adjust in conjunction with other parameters. Weights and logical operators (AND=1) ensure the combined effect of the rules.

[0027] This design dynamically adapts to the system state through fuzzy inference, which can better handle nonlinear and time-varying conditions compared to traditional PID. However, the actual effect depends on the completeness of the rule base and parameter tuning. It needs to be combined with a real-time control loop to convert the fuzzy output into actual PID parameters and verify the dynamic performance.

[0028] Table 3: MATLAB Programming Flowchart

[0029] Table 4: Key Parameters in MATLAB Programming Flow:

[0030] In this embodiment, the programming using an M-file is as follows: %% 1. System Initialization clear all; clc; fis = newfis('NonZeroControl_ZeroIAE', 'mamdani'); 2. Define fuzzy set labels e_mf = {'NB', 'NM', 'NS', 'ZE', 'PS', 'PM', 'PB'}; % Level 7 fuzzy subset 3. Dynamic System Modeling (Dimensional Correction) Ts = 0.001; % 1ms sampling period t = (0:Ts:2)'; % Column vectors ensure consistent dimensions sys = tf([0.02],[1 -1.8 0.82],Ts); ref = ones(size(t)); % Step reference signal 4. Fuzzy System Configuration % Input variable definition fis = addvar(fis, 'input', 'e', ​​[-0.5 0.5]); fis = addvar(fis, 'input', 'de', [-0.3 0.3]); % Output variable definition fis = addvar(fis, 'output', 'delta_Kp', [-0.05 0.05]); fis = addvar(fis, 'output', 'delta_Ki', [-0.02 0.02]); fis = addvar(fis, 'output', 'delta_Kd', [-0.01 0.01]); % Gaussian membership function configuration for i = 1:7 % e's membership function fis = addmf(fis, 'input', 1, ['e_' e_mf{i}], 'gaussmf',... [0.08+0.02*(4-i) -0.5+0.2*(i-1)]); %de's membership function (bandwidth reduction) fis = addmf(fis, 'input', 2, ['de_' e_mf{i}], 'gaussmf',... [(0.08+0.02*(4-i))*0.6 (-0.5+0.2*(i-1))*0.6]); end % Output membership function out_params = [ -0.05, -0.02, -0.01; % NB -0.03, -0.01, -0.005; % NM -0.01, -0.005, -0.002; % NS 0.00, 0.00, 0.00; % ZE 0.01, 0.005, 0.002; % PS 0.03, 0.01, 0.005; % PM 0.05, 0.02, 0.01 % PB ]; for var = 1:3 for i = 1:7 fis = addmf(fis, 'output', var, ['out' num2str(var) '_' e_mf{i}],... 'gaussmf', [0.005+0.002*abs(i-4) out_params(i,var)]); end end 5. Intelligent Rule Base ruleList = [ 4,4,4,4,4,1,1; % Balance rule 3,3,5,3,3,0.8,1; % Negative fine-tuning 5,5,3,5,3,0.8,1; % Positive fine-tuning 2,2,6,2,4,0.6,1; % moderate correction 6,6,2,6,2,0.6,1; 1,1,7,1,5,0.4,1; % Significant correction 7,7,1,7,1,0.4,1; 4,3,4,6,4,0.5,1; % Dynamic compensation 4,5,4,2,4,0.5,1 ]; fis = addrule(fis, ruleList); 6. Control Implementation (Strict Dimensional Management) % Initialize variables (convert all to column vectors) actual = lsim(sys, ref, t); e = ref - actual; de = [0; diff(e) / Ts]; % Ensure column vectors u = zeros(size(t)); % PID parameter initialization Kp = 3.2; Ki = 0.8; Kd = 0.2; delta_Kp = 0; delta_Ki = 0; delta_Kd = 0; for k = 3:length(t) % Error calculation (explicit dimension transformation) e(k) = ref(k) - actual(k); de(k) = (e(k) - e(k-1)) / Ts; % Fuzzy Inference Input Validation norm_e = max(min(2*e(k) / 0.5, 1.2), -1.2); norm_de = max(min(2*de(k) / 0.3, 1.2), -1.2); if abs(norm_e)<=1.2 && abs(norm_de)<=1.2 outputs = evalfis([norm_e, norm_de], fis); else outputs = [0 0 0]; end % Parameter update delta_Kp = 0.4*delta_Kp + 0.6*outputs(1); delta_Ki = 0.4*delta_Ki + 0.6*outputs(2); delta_Kd = 0.4*delta_Kd + 0.6*outputs(3); Kp = max(min(Kp + delta_Kp*0.2, 6.0), 1.0); Ki = max(min(Ki + delta_Ki*0.1, 1.5), 0.1); Kd = max(min(Kd + delta_Kd*0.05, 0.5), 0); % Control quantity generation (vectorized operation) integ_window = max(1,k-500):k;; u(k) = Kp*e(k) + Ki*sum(e(integ_window))*Ts + Kd*de(k) + 0.98*ref(k); % System Update (with Dimensional Inspection) actual(k) = 1.8*actual(k-1) - 0.82*actual(k-2) + 0.02*u(k-1) +0.0001*randn(); end 7. Performance Analysis and Charting figure('Position',[100,100,900,750]); % System Response subplot(3,1,1); plot(t,ref,'r--', t,actual,'b-','LineWidth',1.8); xlabel('Time(s)'); ylabel('Amplitude'); title(['System Response(IAE=' num2str(sum(abs(e))*Ts,'%.2e') ', ISE='num2str(sum(e.^2)*Ts,'%.2e') ')']); legend('reference signal', 'actual output', 'Location', 'southeast'); grid on; ylim([-0.1 1.2]); % Control signal subplot(3,1,2); [ax,h1,h2] = plotyy(t,u,t(1:50:end),[Kp*ones(1,length(t(1:50:end))));... Ki*ones(1,length(t(1:50:end)));... Kd*ones(1,length(t(1:50:end)))]); set(h1,'LineStyle','-','Color','m','LineWidth',2); set(h2(1),'LineStyle',':','Color','r','Marker','.'); set(h2(2),'LineStyle','-.','Color','g','Marker','+'); set(h2(3),'LineStyle','--','Color','b','Marker','x'); xlabel('time(s)'); ylabel(ax(1),'Control Variable'); ylabel(ax(2),'PID Parameter'); legend('Control Variable',['Kp=' num2str(Kp,'%.2f')],['Ki=' num2str(Ki,'%.3f')],['Kd=' num2str(Kd,'%.2f')]); grid on; % Error Analysis subplot(3,1,3); semilogy(t,abs(e)+eps,t,abs(de)+eps,'LineWidth',1.5); xlabel('Time(s)'); ylabel('Error(log)'); title(['Error Analysis(Maximum Error=' num2str(max(abs(e)),'%.2e') ')']); legend('absolute error', 'error rate of change', 'Location', 'northeast'); grid on; ylim([1e-10 1]); In this embodiment, as Figure 1 The results show that the IAE and ISE values ​​are significantly smaller, proving that the hybrid control algorithm proposed in this embodiment, which integrates feedforward compensation and adaptive fuzzy PID, has remarkable effects.

[0031] Unless otherwise specified, in this invention, terms such as "length," "width," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," and "circumferential" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe orientation or positional relationships in this invention are for illustrative purposes only and should not be construed as limiting this application. For those skilled in the art, the specific meaning of the above terms can be understood in conjunction with the accompanying drawings and according to the specific circumstances.

[0032] Unless otherwise explicitly specified and limited, the terms "set up," "connected," and "linked" in this invention should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a direct connection, an indirect connection through an intermediate medium, or a connection within two components. Those skilled in the art can understand the specific meaning of these terms in this invention based on the specific circumstances.

[0033] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any simple modifications or equivalent changes made to the above embodiments based on the technical essence of the present invention shall fall within the protection scope of the present invention.

Claims

1. A hybrid control method integrating feedforward compensation and adaptive fuzzy PID, characterized in that, Includes the following steps: Step 1, Real-time Data Acquisition: Obtain the position error of each axis. and rate of change ; Step 2, Feedforward Compensation Calculation: This includes a feedforward compensation module that generates the feedforward compensation amount through inverse kinematics. ; Step 3, Fuzzy Inference: This includes an adaptive fuzzy PID module that calculates the increments of PID parameters. , , ; Step 4, Prediction Compensation: Establish a multi-axis linkage prediction model, and generate forward prediction compensation based on steps 1, 2, and 3. u comp ; Step 5, control quantity synthesis: in: For adaptive fuzzy PID output, This is the feedforward compensation amount. For forward forecasting compensation; Step 6, Closed-loop execution: Output control input and update system status.

2. The hybrid control method integrating feedforward compensation and adaptive fuzzy PID according to claim 1, characterized in that: The feedforward compensation module is based on multi-axis kinematics inverse analysis to construct an error mapping model, including error inverse derivation and nonlinear compensation model; Error back-end derivation: contour error Decomposed into individual axes: ； Let be the Jacobian matrix, describing the differential relationship between the tool tip error and the errors of each axis: ； in: Generalized coordinate vector It includes linear axes and rotational axes. x, y, z The coordinate vectors of the linear axes. a、c The coordinate vectors of the rotation axis; The tool tip error vector ; Nonlinear compensation model: ； In the formula, Gain compensation for reverse gap; Coulomb friction coefficient; It is a friction speed sensitive factor; This is an estimate of the inter-shaft elastic deformation. It is a hyperbolic tangent function with an output range of (-1, 1) and a smooth S-shaped curve.

3. The hybrid control method integrating feedforward compensation and adaptive fuzzy PID according to claim 1, characterized in that: The adaptive fuzzy PID module is used to acquire position errors in real time. and rate of change It can be adjusted online through 49 fuzzy rules, including fuzzification processing, fuzzy rule base, and parameter updates; Blur processing: Input normalization: ， ； Trigonometric membership functions: In the formula, Center point (NB=-1, PB=1). ; Fuzzy rule base: Typical rule example: IF is NB AND is NB THEN is PB, is NB, is PM; Weighted average solution fuzzy: , ; Parameter update: The update expression is represented as: ； In the formula, the Clip function is mathematically defined as follows: 。 4. The hybrid control method integrating feedforward compensation and adaptive fuzzy PID according to claim 1, characterized in that: Establishing a multi-axis linkage prediction model includes the construction of a state-space prediction model and look-ahead compensation; Construction of the state-space prediction model: The core components are: ； In the formula, , For the system matrix; Inter-axis coupling weights; This is a nonlinear disturbance term; The linear dynamic term describes the error of the system at the previous time step. Impact on the next moment; As a control input, it is used to quantify changes in the control quantity. The ability to correct errors; This refers to inter-axis coupling terms. It is a nonlinear function; q represents the generalized coordinate, a vector describing the position of each axis of the machine tool, including linear axes. X, Y, Z Displacement and rotation axes a、c or b The rotation angle of the axis is expressed as q is a generalized velocity, representing the vector of velocities along each axis; Forward compensation: ； In the formula, To compensate for the gain matrix, a 3-step look-ahead is set.

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