Force control method based on active disturbance rejection control

Abstract

The loading force control method based on active disturbance rejection (ADRR) first establishes a robot kinematic model, including the establishment of the coordinate system of the dynamic and static platforms, the solution of the inverse kinematics of the loading structure, and the derivation of the static mapping relationship between the cylinder output force and the loading force. Secondly, a driver model is established, and a mathematical model of a metal-gap sealed cylinder system driven by a proportional flow valve is performed, deriving an approximate second-order model. Then, an ADRR controller is designed, comprising two modules: transient response and nonlinear feedback. The transient response uses a fastest discrete tracking differentiator to avoid overshoot caused by excessive initial error. The nonlinear feedback design employs a nonlinear feedback control law, combining the error signal, the error derivative signal, and the error integral signal to design an efficient nonlinear combined feedback law to improve control efficiency.

Description

Technical Field

[0001] This invention relates to the field of robotics, and in particular to a loading force control method based on active disturbance rejection control. Background Art

[0002] In today's industrial production and technological development, the application of robotics is becoming increasingly widespread. Precise control and efficient operation of robots are crucial for improving production efficiency, ensuring product quality, and expanding application areas. However, traditional robot control methods often have significant shortcomings when facing complex working conditions and high-precision requirements. For example, in establishing kinematic models, previous methods may not accurately describe the position and orientation of the robot's end effector, limiting the accuracy of motion control. Furthermore, the analysis of loading structures lacks in-depth research, making it difficult to accurately calculate the relationship between cylinder output force and loading force, thus affecting the robot's performance under load.

[0003] In constructing the actuator model, previous mathematical modeling of cylinder systems driven by proportional flow valves was insufficient, failing to fully consider the interaction of various factors such as flow rate, pressure, and dynamics. This resulted in the actuator's control accuracy and response speed failing to meet practical requirements. Regarding controller design, traditional control strategies such as PID control often struggle to achieve ideal control effects for complex nonlinear systems, easily leading to overshoot, oscillations, and other problems that affect the robot's stable operation and work efficiency.

[0004] With the continuous improvement of industrial automation, the performance requirements for robots are becoming increasingly demanding. They not only need to perform simple repetitive actions, but also adapt to complex and changing working environments and high-precision tasks. Therefore, developing a more accurate, efficient, and stable robot control system has become an urgent problem to be solved in the current technological field. Summary of the Invention

[0005] To address the above problems, this invention proposes a loading force control method based on active disturbance rejection control, the specific steps of which are as follows:

[0006] Step 1: Establish the robot's kinematic model; by establishing a coordinate system for the dynamic and static platforms, solve the inverse kinematics of the loading structure, and derive the static mapping relationship between the cylinder output force and the loading force;

[0007] Step 2: Establish the driver model; perform mathematical modeling on the metal gap seal cylinder system driven by the proportional flow valve, and derive an approximate second-order model;

[0008] Step 3: Design an active disturbance rejection controller; for the driver model, design an active disturbance rejection controller, including two modules: transient process and nonlinear feedback;

[0009] Step 4: Transient process design; By using the fastest discrete tracking differentiator, the transient process is arranged to ensure that the controller can quickly track the input signal and acquire its differential signal, avoiding overshoot problems caused by excessive initial error;

[0010] Step 5: Nonlinear feedback design; Using a nonlinear feedback control law, combined with the error signal, error derivative signal, and error integral signal, an efficient nonlinear combined feedback law is designed to improve the control efficiency of the controller.

[0011] Furthermore, the process of establishing the robot's kinematic model in step 1 can be represented as:

[0012] Step 1.1 Establish the coordinate system of the dynamic and static platforms

[0013] Step 1.1.1 Static Platform Coordinate System: Establish the Base Coordinate System

[0014] Static platform coordinate system It is a fixed reference coordinate system, with the robot base as the coordinate system. ;make The origin is The basis vectors are ;

[0015] in, Perpendicular to the base plane, and Within the plane of the base;

[0016] Step 1.1.2 Moving Platform Coordinate System: Establish the End Coordinate System

[0017] Moving platform coordinate system It is a coordinate system that moves with the robot's end effector; let The origin is The basis vectors are ;in, Perpendicular to the plane of the end effector and Within the plane of the end effector;

[0018] Step 1.1.3 Coordinate Transformation

[0019] Moving platform coordinate system Relative to the static platform coordinate system Position and orientation are determined by homogeneous transformation matrix express:

[0020]

[0021] in: It is a 3×3 rotation matrix, representing Compared to The posture; It is a 3×1 translation vector, representing Compared to Location;

[0022] Step 1.1.4 Derive the rotation matrix :

[0023] The rotation matrix is ​​calculated using Euler angles, where Euler angles are... Then the rotation matrix is:

[0024]

[0025] in, It is a rotation matrix about the z-axis. It is a rotation matrix about the y-axis. It is a rotation matrix about the x-axis:

[0026]

[0027]

[0028]

[0029] Step 1.2 Solve the inverse kinematics of the loaded structure.

[0030] Step 1.2.1 The goal of inverse kinematics:

[0031] The position of the end effector is known. and posture Solve for the displacement vector of each joint. , It is the displacement variable of the nth joint;

[0032] Step 1.2.2 Numerical Solution

[0033] For complex robot structures, the nonlinear equations are solved numerically; let the kinematic equations be... Solve iteratively:

[0034]

[0035] in, yes The displacement variables of the joint, yes The displacement variables of the joint, It is a Jacobian matrix, defined as:

[0036]

[0037] in, These are partial derivatives;

[0038] Step 1.3 Derive the static mapping relationship between cylinder output force and loading force.

[0039] Step 1.3.1 Loading Force Analysis

[0040] The loading force on the end effector is , yes The component in the x-axis direction, yes The component in the y-axis direction, yes The component in the z-axis direction is used to establish the relationship between the cylinder output force and the loading force through static equilibrium conditions;

[0041] Step 1.3.2 Static Mapping Relationship

[0042] The force vector of the cylinder is , It is the output force of the nth cylinder, and the loading force vector is... By using the static equilibrium equations, a mapping relationship is established:

[0043]

[0044] in It is the Jacobian matrix, representing the geometric relationship between cylinder force and loading force;

[0045] Step 1.3.3 Derivation of the Jacobian Matrix

[0046] Jacobian matrix Calculated using the position vector of the platform hinge point and the cylinder direction vector;

[0047] Cylinder direction vector is The hinge point position vector is Then the Jacobian matrix of the first degree Listed as:

[0048]

[0049] in Represents vector product.

[0050] Furthermore, the driver model established in step 2 can be represented as follows:

[0051] Step 2.1: Establish the relationship between flow rate and pressure:

[0052] A proportional flow valve regulates its output flow rate by controlling an input electrical signal. For a proportional flow valve, its flow rate... With input control signal and the pressure difference across the valve port Related, expressed as:

[0053]

[0054] in, For flow coefficient; It is the valve port area, relative to the input signal. Proportional relationship; It is the fluid density;

[0055] Let the pressure in the rodless chamber of the cylinder be The pressure in the rod chamber is When the proportional flow valve controls the flow rate entering the rodless chamber of the cylinder:

[0056]

[0057] in, It is the pressure difference across the valve port. For oil supply pressure;

[0058] Step 2.2: Cylinder dynamics analysis:

[0059] In a cylinder system, the resultant force acting on the piston For cylinder output force With load force The difference; as can be seen from step 1, the cylinder output force:

[0060]

[0061] in, The area of ​​the rodless chamber piston is... The piston area of ​​the rod chamber;

[0062] Piston acceleration With displacement The second derivative The relationship is represented as:

[0063]

[0064] load capacity Including friction Inertial force In addition to external loading forces, friction force, etc. It is approximately the sum of Coulomb friction and viscous friction, that is:

[0065]

[0066] in, Coulomb friction, The coefficient of viscous friction is... displacement The first derivative;

[0067] Step 2.3: Establish the gas law:

[0068] The gas inside the cylinder is considered an ideal gas, obeying the ideal gas law. During the cylinder's movement, the gas volume... With piston displacement Changes in the volume of the rodless cavity :

[0069]

[0070] in, Given the initial volume, the volume of the rod cavity is... :

[0071]

[0072] in, Given the initial volume, we can obtain:

[0073]

[0074] in, The adiabatic index, , It is a constant;

[0075] Step 2.4: Derive the approximate second-order model:

[0076] Solving the flow equation, cylinder dynamics equation, and gas state equation simultaneously yields a solution for the piston displacement. Approximate second-order linear differential equation:

[0077]

[0078] in, The equivalent mass of the piston and load. The coefficient of viscous friction is... The equivalent spring stiffness of the cylinder. This is the control coefficient for the proportional flow valve. External loading force coefficient, Input control signal to proportional flow valve. The external loading force is represented by the equation, which is an approximate second-order model of the metal gap seal cylinder system driven by the proportional flow valve.

[0079] Furthermore, the transition process design in step 4 can be represented as follows:

[0080] Step 4.1: Establish the fastest discrete tracking differentiator

[0081] Let the desired input signal of the system be... , The time variable, after discretization, becomes , Given discrete-time variables, two sequences are obtained using the fastest discrete tracking differentiator. and ;in Used for rapid tracking of input signals ,and Approximately The differential signal;

[0082] Step 4.2: The iterative formula for the fastest discrete tracking differentiator is as follows:

[0083]

[0084] in, Indicates the first Rapidly track the input signal, Indicates the first of differential signal, Represents the discrete time step; It is the tracking speed parameter; The larger, track The faster the speed, the more drastic the system response may become, or even overshoot. The smaller the value, the slower the tracking speed, but the smoother the system response. (This refers to the tracking speed parameter of the differentiator.) and discrete time step It can be adjusted based on the dynamic characteristics of the system:

[0085]

[0086] in, It is the equivalent mass of the piston and load in the approximate second-order model. It is the equivalent spring stiffness of the cylinder. It is the system damping ratio:

[0087]

[0088] in, It is the viscous friction coefficient in the approximate second-order model, a function. It is the core component of the fastest discrete tracking differentiator, and its definition is:

[0089] in, , yes variable, , It is a symbolic function.

[0090] Furthermore, the nonlinear feedback design in step 5 is expressed as follows:

[0091] Step 5.1: Define the error signal

[0092] Input signal The fastest discrete tracking differentiator generates a smooth reference signal. and its differential signal ;

[0093] The actual output of the system is Then the error signal Defined as:

[0094]

[0095] For error signals Differentiate to obtain the error differential signal :

[0096]

[0097] For error signals Integrate to obtain the error integral signal. ;

[0098] Step 5.2: Design of Nonlinear Feedback Control Law

[0099] The nonlinear feedback control law takes the following form:

[0100]

[0101] in, It is the output of the controller, i.e., the control quantity; , , These are the proportional coefficient, differential coefficient, and integral coefficient, respectively. , , It is a nonlinear function relating to the error signal, the differential error signal, and the integral error signal;

[0102] Step 5.3: Description of Nonlinear Functions

[0103] The nonlinear functions of the error signal, the error derivative signal, and the error integral signal all employ saturation functions, which are defined as follows:

[0104]

[0105] in, It is the saturation value. It is a nonlinear function variable; when the error signal or its derivative or integral is large, the saturation function can limit the magnitude of the control quantity, avoiding excessive overshoot or oscillation in the system. The loading force control method based on active disturbance rejection control in this invention has the following beneficial effects: The technical effects of this invention are:

[0106] 1. The comprehensive actuator model of this invention provides a solid foundation for precise actuator control. In-depth mathematical modeling of the proportional flow valve-driven metal-gap sealed cylinder system enables more precise actuator control, faster response speed, and effectively reduces system latency and errors.

[0107] 2. The design of the active disturbance rejection controller of the present invention, especially the combination of the transient process and the nonlinear feedback module, can effectively avoid the overshoot problem caused by excessive initial error and enhance the stability and robustness of the system.

[0108] 3. The application of the nonlinear feedback control law in this invention, combined with the error signal, the error differential signal, and the error integral signal, significantly improves the control efficiency of the controller, enabling the robot to respond quickly and accurately when facing complex and ever-changing working environments and tasks, thereby improving work efficiency and quality. Attached Figure Description

[0109] Figure 1 This is a flowchart of the present invention.

[0110] Figure 2 This is a schematic diagram illustrating the establishment of the coordinate system in this invention. Detailed Implementation

[0111] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0112] This invention relates to robot control technology, including establishing a robot kinematic model, a actuator model, and an active disturbance rejection controller. Through precise modeling and innovative control strategies, it addresses the shortcomings of traditional control methods. This improves control accuracy, response speed, and stability, enhancing work efficiency and quality, and adapting to complex working conditions and high-precision task requirements. The invention flowchart is shown below. Figure 1 As shown, the steps of the present invention will be described in detail below.

[0113] Step 1: Establish the robot's kinematic model. By establishing a coordinate system for the dynamic and static platforms, solve the inverse kinematics of the loading structure and derive the static mapping relationship between the cylinder output force and the loading force.

[0114] Step 1.1 Establish the coordinate system of the dynamic and static platforms

[0115] Step 1.1.1 Static Platform Coordinate System: Establish the Base Coordinate System

[0116] Static platform coordinate system It is a fixed reference coordinate system, with the robot base as the coordinate system. .make The origin is The basis vectors are .

[0117] in, Perpendicular to the base plane, and A schematic diagram of the coordinate system establishment within the base plane is shown below. Figure 2 As shown.

[0118] Step 1.1.2 Moving Platform Coordinate System: Establish the End Coordinate System

[0119] Moving platform coordinate system It is a coordinate system that moves with the robot's end effector. Let... The origin is The basis vectors are .in, Perpendicular to the plane of the end effector and Within the end effector plane.

[0120] Step 1.1.3 Coordinate Transformation

[0121] Moving platform coordinate system Relative to the static platform coordinate system Position and orientation are determined by homogeneous transformation matrix express:

[0122]

[0123] in: It is a 3×3 rotation matrix, representing Compared to The posture. It is a 3×1 translation vector, representing Compared to The location.

[0124] Step 1.1.4 Derive the rotation matrix :

[0125] The rotation matrix is ​​calculated using Euler angles, where Euler angles are... Then the rotation matrix is:

[0126]

[0127] in, It is a rotation matrix about the z-axis. It is a rotation matrix about the y-axis. It is a rotation matrix about the x-axis:

[0128]

[0129]

[0130]

[0131] Step 1.2 Solve the inverse kinematics of the loaded structure.

[0132] Step 1.2.1 The goal of inverse kinematics:

[0133] The position of the end effector is known. and posture Solve for the displacement vector of each joint. , It is the displacement variable of the nth joint.

[0134] Step 1.2.2 Numerical Solution

[0135] For complex robot structures, a numerical method is used to solve the nonlinear equations. Let the kinematic equations be... Solve iteratively:

[0136]

[0137] in, yes The displacement variables of the joint, yes The displacement variables of the joint, It is a Jacobian matrix, defined as:

[0138]

[0139] in, It is a partial derivative.

[0140] Step 1.3 Derive the static mapping relationship between cylinder output force and loading force.

[0141] Step 1.3.1 Loading Force Analysis

[0142] The loading force on the end effector is , yes The component in the x-axis direction, yes The component in the y-axis direction, yes The component in the z-axis direction is used to establish the relationship between the cylinder output force and the loading force through static equilibrium conditions.

[0143] Step 1.3.2 Static Mapping Relationship

[0144] The force vector of the cylinder is , It is the output force of the nth cylinder, and the loading force vector is... By using the static equilibrium equations, a mapping relationship is established:

[0145]

[0146] in It is the Jacobian matrix, representing the geometric relationship between cylinder force and loading force.

[0147] Step 1.3.3 Derivation of the Jacobian Matrix

[0148] Jacobian matrix Calculated using the position vector of the platform hinge point and the cylinder direction vector.

[0149] Cylinder direction vector is The hinge point position vector is Then the Jacobian matrix of the first degree Listed as:

[0150]

[0151] in Represents vector product.

[0152] Step 2: Establish the driver model. A mathematical model is performed on the proportional flow valve-driven metal-gap sealed cylinder system, deriving an approximate second-order model.

[0153] Step 2.1: Establish the relationship between flow rate and pressure:

[0154] A proportional flow valve regulates its output flow rate by controlling an input electrical signal. For a proportional flow valve, its flow rate... With input control signal and the pressure difference across the valve port Related, expressed as:

[0155]

[0156] in, For flow coefficient; It is the valve port area, relative to the input signal. Proportional relationship; It is the fluid density.

[0157] Let the pressure in the rodless chamber of the cylinder be The pressure in the rod chamber is When the proportional flow valve controls the flow rate entering the rodless chamber of the cylinder:

[0158]

[0159] in, It is the pressure difference across the valve port. This refers to the oil supply pressure.

[0160] Step 2.2: Cylinder dynamics analysis:

[0161] In a cylinder system, the resultant force acting on the piston For cylinder output force With load force The difference. From step 1, we know that the cylinder output force is:

[0162]

[0163] in, The area of ​​the rodless chamber piston is... This represents the piston area in the rod chamber.

[0164] Piston acceleration With displacement The second derivative The relationship is represented as:

[0165]

[0166] load capacity Including friction Inertial force In addition to external loading forces, friction force, etc. It is approximately the sum of Coulomb friction and viscous friction, that is:

[0167]

[0168] in, Coulomb friction, The coefficient of viscous friction is... displacement The first derivative.

[0169] Step 2.3: Establish the gas law:

[0170] The gas inside the cylinder is considered an ideal gas, obeying the ideal gas law. During the cylinder's movement, the gas volume... With piston displacement Changes in the volume of the rodless cavity :

[0171]

[0172] in, Given the initial volume, the volume of the rod cavity is... :

[0173]

[0174] in, Given the initial volume, we can obtain:

[0175]

[0176] in, The adiabatic index, , It is a constant.

[0177] Step 2.4: Derive the approximate second-order model:

[0178] Solving the flow equation, cylinder dynamics equation, and gas state equation simultaneously yields a solution for the piston displacement. Approximate second-order linear differential equation:

[0179]

[0180] in, The equivalent mass of the piston and load. The coefficient of viscous friction is... The equivalent spring stiffness of the cylinder. This is the control coefficient for the proportional flow valve. External loading force coefficient, Input control signal to proportional flow valve. The external loading force is represented by this equation, which is an approximate second-order model of the metal gap seal cylinder system driven by the proportional flow valve.

[0181] Step 3: Design an active disturbance rejection controller. For the driver model, design an active disturbance rejection controller, including two modules: transient process and nonlinear feedback.

[0182] Step 4: Transient Process Design. By using the fastest discrete tracking differentiator, a transient process is designed to ensure that the controller can quickly track the input signal and acquire its differential signal, avoiding overshoot problems caused by excessive initial error.

[0183] Step 4.1: Establish the fastest discrete tracking differentiator

[0184] Let the desired input signal of the system be... , The time variable, after discretization, becomes , Given discrete-time variables, two sequences are obtained using the fastest discrete tracking differentiator. and .in Used for rapid tracking of input signals ,and Approximately The differential signal.

[0185] Step 4.2: The iterative formula for the fastest discrete tracking differentiator is as follows:

[0186]

[0187] in, Indicates the first Rapidly track the input signal, Indicates the first of differential signal, This represents the discrete time step. It is the tracking speed parameter. The larger, track The faster the speed, the more drastic the system response may become, or even overshoot. The smaller the value, the slower the tracking speed, but the smoother the system response. (This refers to the tracking speed parameter of the differentiator.) and discrete time step It can be adjusted based on the dynamic characteristics of the system:

[0188]

[0189] in, It is the equivalent mass of the piston and load in the approximate second-order model. It is the equivalent spring stiffness of the cylinder. It is the system damping ratio:

[0190]

[0191] in, It is the viscous friction coefficient in the approximate second-order model, a function. It is the core component of the fastest discrete tracking differentiator, and its definition is:

[0192] in, , yes variable, , It is a symbolic function.

[0193] Step 5: Nonlinear Feedback Design. A nonlinear feedback control law is employed, combining the error signal, the error derivative signal, and the error integral signal to design an efficient nonlinear combined feedback law, thereby improving the controller's control efficiency.

[0194] Step 5.1: Define the error signal

[0195] Input signal The fastest discrete tracking differentiator generates a smooth reference signal. and its differential signal .

[0196] The actual output of the system is Then the error signal Defined as:

[0197]

[0198] For error signals Differentiate to obtain the error differential signal :

[0199]

[0200] For error signals Integrate to obtain the error integral signal. .

[0201] Step 5.2: Design of Nonlinear Feedback Control Law

[0202] The nonlinear feedback control law takes the following form:

[0203]

[0204] in, It is the output of the controller, i.e., the control quantity; , , These are the proportional coefficient, differential coefficient, and integral coefficient, respectively. , , It is a nonlinear function relating to the error signal, the differential error signal, and the integral error signal.

[0205] Step 5.3: Description of Nonlinear Functions

[0206] The nonlinear functions of the error signal, the error derivative signal, and the error integral signal all employ saturation functions, which are defined as follows:

[0207]

[0208] in, It is the saturation value. It is a nonlinear function variable. When the error signal or its derivative or integral is large, the saturation function can limit the magnitude of the control quantity and avoid excessive overshoot or oscillation in the system.

[0209] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any modifications or equivalent changes made based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A loading force control method based on active disturbance rejection control, comprising the following specific steps, characterized in that: Step 1: Establish the robot's kinematic model; by establishing a coordinate system for the dynamic and static platforms, solve the inverse kinematics of the loading structure, and derive the static mapping relationship between the cylinder output force and the loading force; Step 1.1 Establish the coordinate system of the static and dynamic platforms; Step 1.2 Solve the inverse kinematics of the loaded structure. Step 1.2.1 The goal of inverse kinematics: The position of the end effector is known. and posture Solve for the displacement vector of each joint. , It is the displacement variable of the nth joint; Step 1.2.2 Numerical Solution For complex robot structures, the nonlinear equations are solved numerically; let the kinematic equations be... Solve iteratively: in, yes The displacement variables of the joint, yes The displacement variables of the joint, It is a Jacobian matrix, defined as: in, These are partial derivatives; Step 1.3 Derive the static mapping relationship between cylinder output force and loading force. Step 1.3.1 Loading Force Analysis The loading force on the end effector is , yes The component in the x-axis direction, yes The component in the y-axis direction, yes The component in the z-axis direction is used to establish the relationship between the cylinder output force and the loading force through static equilibrium conditions; Step 1.3.2 Static Mapping Relationship The force vector of the cylinder is , It is the output force of the nth cylinder, and the loading force vector is... By using the static equilibrium equations, a mapping relationship is established: in It is the Jacobian matrix, representing the geometric relationship between cylinder force and loading force; Step 1.3.3 Derivation of the Jacobian Matrix Jacobian matrix Calculated using the position vector of the platform hinge point and the cylinder direction vector; Cylinder direction vector is The hinge point position vector is Then the Jacobian matrix of the first degree Listed as: in Represents vector product; Step 2: Establish the driver model; perform mathematical modeling on the metal gap seal cylinder system driven by the proportional flow valve, and derive an approximate second-order model; The driver model established in step 2 can be represented as follows: Step 2.1: Establish the relationship between flow rate and pressure: A proportional flow valve regulates its output flow rate by controlling an input electrical signal. For a proportional flow valve, its flow rate... With input control signal and the pressure difference across the valve port Related, expressed as: in, For flow coefficient; It is the valve port area, relative to the input signal. Proportional; It is the fluid density; Let the pressure in the rodless chamber of the cylinder be The pressure in the rod chamber is When the proportional flow valve controls the flow rate entering the rodless chamber of the cylinder: in, It is the pressure difference across the valve port. For oil supply pressure; Step 2.2: Cylinder dynamics analysis: In a cylinder system, the resultant force acting on the piston For cylinder output force With load force The difference; as can be seen from step 1, the cylinder output force: in, The area of ​​the rodless chamber piston is... The piston area of ​​the rod chamber; Piston acceleration With displacement The second derivative The relationship is represented as: load capacity Including friction Inertial force In addition to external loading forces, friction force, etc. It is approximately the sum of Coulomb friction and viscous friction, that is: in, Coulomb friction, The coefficient of viscous friction is... displacement The first derivative; Step 2.3: Establish the gas law: The gas inside the cylinder is considered an ideal gas, obeying the ideal gas law. During the cylinder's movement, the gas volume... With piston displacement Changes in the volume of the rodless cavity : in, Given the initial volume, the volume of the rod cavity is... : in, Given the initial volume, we can obtain: in, The adiabatic index, , It is a constant; Step 2.4: Derive the approximate second-order model: Solving the flow equation, cylinder dynamics equation, and gas state equation simultaneously yields a solution for the piston displacement. Approximate second-order linear differential equation: in, The equivalent mass of the piston and load. The coefficient of viscous friction is... The equivalent spring stiffness of the cylinder. This is the control coefficient for the proportional flow valve. External loading force coefficient, Input control signal to the proportional flow valve. The external loading force is represented by this equation, which is an approximate second-order model of the metal gap seal cylinder system driven by the proportional flow valve. Step 3: Design an active disturbance rejection controller; for the driver model, design an active disturbance rejection controller, including two modules: transient process and nonlinear feedback; Step 4: Transient process design; By using the fastest discrete tracking differentiator, the transient process is arranged to ensure that the controller can quickly track the input signal and acquire its differential signal, avoiding overshoot problems caused by excessive initial error; The transition process design in step 4 can be represented as follows: Step 4.1: Establish the fastest discrete tracking differentiator Let the desired input signal of the system be... , The time variable, after discretization, becomes , Given discrete-time variables, two sequences are obtained using the fastest discrete tracking differentiator. and ;in Used for rapid tracking of input signals ,and Approximately The differential signal; Step 4.2: The iterative formula for the fastest discrete tracking differentiator is as follows: in, Indicates the first Rapidly track the input signal, Indicates the first of differential signal, Represents the discrete time step; It is the tracking speed parameter; The larger, track The faster the speed, the more drastic the system response may become, or even overshoot. The smaller the value, the slower the tracking speed, but the smoother the system response. (This refers to the tracking speed parameter of the differentiator.) and discrete time step It can be adjusted based on the dynamic characteristics of the system: in, It is the equivalent mass of the piston and load in the approximate second-order model. It is the equivalent spring stiffness of the cylinder. It is the system damping ratio: in, It is the viscous friction coefficient in the approximate second-order model, a function. It is the core component of the fastest discrete tracking differentiator, and its definition is: in, , yes variable, , For symbolic functions Step 5: Nonlinear feedback design; Using a nonlinear feedback control law, combined with the error signal, the error derivative signal, and the error integral signal, an efficient nonlinear combined feedback law is designed to improve the control efficiency of the controller. The nonlinear feedback design in step 5 is represented as follows: Step 5.1: Define the error signal Input signal The fastest discrete tracking differentiator generates a smooth reference signal. and its differential signal ; The actual output of the system is Then the error signal Defined as: For error signals Differentiate to obtain the error differential signal : For error signals Integrate to obtain the error integral signal. ; Step 5.2: Design of Nonlinear Feedback Control Law The nonlinear feedback control law takes the following form: in, It is the output of the controller, i.e., the control quantity; , , These are the proportional coefficient, differential coefficient, and integral coefficient, respectively. , , It is a nonlinear function relating to the error signal, the differential error signal, and the integral error signal; Step 5.3: Description of Nonlinear Functions The nonlinear functions of the error signal, the error derivative signal, and the error integral signal all employ saturation functions, which are defined as follows: in, It is the saturation value. It is a nonlinear function variable; when the error signal or its derivative or integral is large, the saturation function can limit the magnitude of the control quantity and avoid excessive overshoot or oscillation in the system.

2. The loading force control method based on active disturbance rejection control according to claim 1, characterized in that: Step 1.1 Establish the coordinate system of the dynamic and static platforms as follows: Step 1.1.1 Static Platform Coordinate System: Establish the Base Coordinate System Static platform coordinate system It is a fixed reference coordinate system, with the robot base as the coordinate system. ;make The origin is The basis vectors are ; in, Perpendicular to the base plane, and Within the plane of the base; Step 1.1.2 Moving Platform Coordinate System: Establish the End Coordinate System Moving platform coordinate system It is a coordinate system that moves with the robot's end effector; let The origin is The basis vectors are ;in, Perpendicular to the plane of the end effector and Within the plane of the end effector; Step 1.1.3 Coordinate Transformation Moving platform coordinate system Relative to the static platform coordinate system Position and orientation are determined by homogeneous transformation matrix express: in: It is a 3×3 rotation matrix, representing Compared to The posture; It is a 3×1 translation vector, representing Compared to Location; Step 1.1.4 Derive the rotation matrix : The rotation matrix is ​​calculated using Euler angles, where Euler angles are... Then the rotation matrix is: in, It is a rotation matrix about the z-axis. It is a rotation matrix about the y-axis. It is a rotation matrix about the x-axis: 。

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