Hydraulic mechanical arm nonlinear motion control method based on dynamic friction accurate compensation

Abstract

This invention discloses a nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation. The method includes: establishing a system dynamic parameterization and improved dynamic friction model for the multi-joint hydraulic robotic arm under dynamic model constraints; designing an improved LuGre friction model compensation term; designing an adaptive robust controller, inputting the reference trajectory into the controller and the compensation term, the controller outputting a dynamic parameter estimation matrix into the compensation term, the compensation term outputting a friction force compensation term into the controller, the controller outputting the valve control voltage to the robotic arm for control, and the robotic arm providing feedback on its state, thereby achieving nonlinear motion control of the robotic arm. This invention enables real-time estimation of the low-speed dynamic friction force of a multi-joint hydraulic robotic arm, solving the problem of difficulty in describing the nonlinear dynamics of friction force in actual low-speed robotic arm operation. While ensuring the overall stability of the control system, it reduces the tracking error of the robotic arm's end effector and improves control performance.

Description

Technical Field

[0001] This invention relates to a nonlinear motion control method for a hydraulic robotic arm, specifically to a nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation. Background Technology

[0002] Hydraulic robotic arms are typically used for demanding tasks such as heavy-duty operations. However, with industrial development and continuous advancements in human exploration, the precision of hydraulic robotic arms has become crucial. Traditional open-loop control and proportional-integral-differential (PID) control, failing to consider factors such as the uncertainty of the robotic arm's dynamic model parameters, are increasingly unable to meet the demands for high control performance. In this context, developing a nonlinear controller based on the dynamic model of a multi-joint hydraulic robotic arm is an effective solution. However, due to the highly nonlinear relationship between friction and velocity, insufficient control of the hydraulic actuator during the start-stop and low-speed movement phases of the hydraulic joints can lead to undesirable crawling or oscillating behaviors, ultimately affecting the tracking performance of the end effector. Furthermore, due to the multi-degree-of-freedom motion characteristics of the robotic arm, even during high-speed movement of the end effector, some joints remain in a low-speed state, causing crawling and oscillating behaviors to persist, limiting the control performance of multi-joint hydraulic robotic arms. In most existing hydraulic actuator control designs, friction is simply described as a combination of Coulomb friction and viscous friction. Therefore, existing controllers struggle to precisely control the joints of hydraulic robotic arms moving at low speeds, making it difficult to guarantee good end-effector control accuracy and affecting operational performance. Summary of the Invention

[0003] To address the problems existing in the background technology, this invention provides a nonlinear motion control method for hydraulic robotic arms based on precise dynamic friction compensation. Specifically, this invention is a motion control method for low-speed motion conditions frequently exhibited during the movement of multi-joint hydraulic robotic arms. This method improves the control accuracy of multi-joint hydraulic robotic arms under conditions of strong nonlinear friction during low-speed movement, achieving precise compensation control for nonlinear friction. While ensuring the stability of the control system, it reduces the end-effector tracking error of the multi-joint hydraulic robotic arm, alleviates crawling and oscillation phenomena, enhances control performance, and improves the control accuracy of the end effector, thereby improving the working performance of the robotic arm in more demanding operating environments.

[0004] The technical solution adopted in this invention is:

[0005] The present invention provides a nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation, comprising:

[0006] Step 1: Taking into account the high-order, multi-degree-of-freedom coupling and nonlinear friction characteristics of the hydraulic manipulator, establish a system dynamic parameterization and dynamic friction improvement model of the multi-joint hydraulic manipulator under the constraints of the dynamic model.

[0007] Step 2: Based on the dynamic friction dynamics characteristics, according to the system dynamics parameterization and the dynamic friction improvement model, the improved LuGre friction model compensation term is designed using the expected compensation method. The improved LuGre friction model compensation term can obtain the nominal value of the internal friction state, realize the real-time update of the unmeasurable internal friction state and the real-time estimation of the friction compensation term.

[0008] Step 3: Design an adaptive robust controller for a multi-joint hydraulic manipulator based on nonlinear dynamic friction compensation. Input the reference trajectory of the multi-joint hydraulic manipulator into the adaptive robust controller and the improved LuGre friction model compensation term. The adaptive robust controller outputs the estimated value of the dynamic parameter matrix of the multi-joint hydraulic manipulator into the improved LuGre friction model compensation term. The improved LuGre friction model compensation term outputs the friction force compensation term into the adaptive robust controller. The adaptive robust controller outputs the control voltage of the valve of the multi-joint hydraulic manipulator to control the multi-joint hydraulic manipulator. Input the state feedback quantity of the multi-joint hydraulic manipulator during motion into the improved LuGre friction model compensation term and the adaptive robust controller of the multi-joint hydraulic manipulator to complete the closed loop and realize the nonlinear motion control of the multi-joint hydraulic manipulator.

[0009] In the first step, the system dynamics parameterization and dynamic friction improvement model of the multi-joint hydraulic manipulator are as follows:

[0010]

[0011]

[0012]

[0013]

[0014] in, , and Let these represent the inertial dynamics matrix, Coriolis force matrix, and gravity matrix of the multi-joint hydraulic manipulator, respectively. ; and These represent the joint angular velocity and joint angular acceleration of a multi-joint hydraulic robotic arm, respectively. This indicates the joint angles of a multi-joint hydraulic robotic arm. ; Denotes a non-singular joint Jacobian matrix. ; and Let represent the equivalent thrust and its derivative of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm. ; Represents frictional torque. , , This represents the first linear regression matrix. Denotes the first dynamic parameter matrix. , , , , and These represent the first, second, third, fourth, and fifth dynamic parameters, respectively, which represent stiffness, damping coefficient, Coulomb friction coefficient, viscous friction coefficient, and the bounded deviation of the first lumped dynamics. Indicates the total modeling error of the first set. The bounded deviation value, the first lumped modeling error Including external disturbances and other terms that are difficult to model, ; , and These represent the sixth, seventh, and eighth dynamic parameters, respectively. , This represents the second dynamic parameter matrix. and These represent the bounded deviations in flow rate for rodless cavity dynamics and rod cavity dynamics, respectively. , Indicates the effective bulk modulus of hydraulic oil; , and These represent the first, second, and third inherent hardware parameters, respectively. , , , and These represent the contact areas of the oil inlet and return chambers of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and These represent the total compressible volumes of the oil inlet and return chambers of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm, respectively. This indicates the control flow rate of the valve in a multi-joint hydraulic robotic arm. , and These represent the actual flow rates of the inlet and outlet oil chambers of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm, respectively. Indicates the total modeling error of the second set. The bounded deviation value, , and Let represent the bounded deviation of flow rate in the rodless cavity dynamics and the rod cavity dynamics of the multi-joint hydraulic manipulator, respectively. ; Let represent the parameter set of the j-th dynamic parameter at each joint of a multi-joint hydraulic robotic arm. and These represent the dynamic parameters on the first joint and its j-th element, respectively. and Let i and j represent the dynamic parameters of the i-th joint and their j-th elements, respectively. and These represent the dynamic parameters of the nth joint and its j-th element, respectively. and These represent the dynamic parameters on the i-th joint. Stiffness and damping coefficient in and These represent the sets of stiffness and damping coefficients for each joint. ; and These represent the dynamic parameters on the i-th joint. The Coulomb coefficient of friction and the viscous coefficient of friction in the figure. and Let these represent the sets of Coulomb friction coefficients and viscous friction coefficients for each joint. ; , and These represent the dynamic parameters on the i-th joint. The first ensemble modeling error The nominal value and the bounded deviation of the flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The nominal value; This represents the switching factor; when the angular velocity exceeds a critical value proportional to the sampling rate, Can be avoided Instability in digital implementation; and Let them represent the internal friction state and its derivative, respectively. , Represents a diagonal matrix. These represent the internal friction states of the 1st, 2nd, ..., nth joints, respectively. This represents the decay function that decays in fractional form. This represents the damping coefficient related to velocity; Represents the identity matrix; Represents the differentiability parameter matrix, This can ensure frictional torque Differentiability; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. a diagonal matrix; This represents the positive matrix of the Stribeck effect.

[0015] In the first step described above, the constraints of the dynamic model are as follows:

[0016]

[0017]

[0018]

[0019]

[0020] in, The inertial dynamics matrix of a multi-joint hydraulic robotic arm The derivative, and These represent the inertial dynamics matrices of the multi-joint hydraulic robotic arm. The minimum and maximum eigenvalues; This indicates the driving torque of each joint in a multi-joint hydraulic robotic arm. ; and These represent the oil pressure in the inlet and outlet chambers of each hydraulic cylinder. ; This represents the matrix of the i-th dynamic parameters. Denotes the physically feasible set of the i-th dynamic parameter matrix. and Represent the i-th dynamic parameter matrix respectively The minimum and maximum values; , and These represent the total modeling error of the first set. bounded deviation value Bounded deviation of flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The threshold; , , , and All of them are known scalars.

[0021] In the second step, the compensation term of the improved LuGre friction model is as follows:

[0022]

[0023]

[0024]

[0025]

[0026]

[0027]

[0028] in, Represents frictional torque The estimated value is the friction compensation term. This represents the expected first linear regression matrix. This represents the estimated value of the first dynamic parameter matrix; and Representing the desired internal friction state and its derivative, respectively, can be expressed as: and ; Represents the zero vector; This represents an estimate of the frictional torque. and frictional torque The error between them Represents the estimated value of the first dynamic parameter matrix and the first dynamic parameter matrix The error between; This represents the residual term in the friction estimate; and These represent the internal friction states respectively. and its derivative The explicit expression, i.e., the internal friction state and its derivative The solution, and , and These represent the first and second parameters within the preset feasible set of angular velocities, respectively. This represents the desired joint angular velocity of a multi-joint hydraulic robotic arm. a diagonal matrix; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angular velocity Tracking error between A diagonal matrix.

[0029] The improved LuGre friction model uses the idea of ​​expected compensation in its compensation terms, based on the derivative of the expected internal friction state. Continuously update the internal friction state .

[0030] The reference trajectory of the multi-joint hydraulic manipulator, with the input of the improved LuGre friction model compensation term, is the desired joint angular velocity of the multi-joint hydraulic manipulator. The estimated value of the dynamic parameter matrix of the multi-joint hydraulic manipulator is the estimated value of the first dynamic parameter matrix. .

[0031] In the third step, the adaptive robust controller of the multi-joint hydraulic manipulator includes a virtual control thrust control law based on a first robustness constraint, a desired control flow control law based on a second robustness constraint, a valve orifice flow mapping function, and an online parameter adaptive law. The reference trajectory of the multi-joint hydraulic manipulator is input into the virtual control thrust control law. The online parameter adaptive law inputs the estimated values ​​of the dynamically updated dynamic parameter matrix into the improved LuGre friction model compensation term, the virtual control thrust control law, and the desired control flow control law, respectively. The improved LuGre friction model compensation term outputs a friction compensation term into the virtual control thrust control law. The virtual control thrust control law outputs virtual control thrust into the desired control flow control law. The desired control flow control law outputs the desired control flow into the valve orifice flow mapping function. The valve orifice flow mapping function outputs the control voltage of the valve of the multi-joint hydraulic manipulator to control the multi-joint hydraulic manipulator.

[0032] The virtual control thrust control law based on the first robustness constraint is as follows:

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039] in, This represents the virtual control thrust of a multi-joint hydraulic robotic arm. and These represent the first adaptive model compensation control law and the first robust feedback control law, respectively. and These represent the first dynamic compensation control law and the first fast dynamic error compensation control law, respectively. and These represent the first linear feedback control law and the first nonlinear robust feedback control law, respectively. and These represent the transition joint angular velocity and transition joint angular acceleration of the multi-joint hydraulic robotic arm, respectively. This indicates the friction compensation term; Represents the first dynamic parameter matrix The fifth dynamic parameter in The estimated value; The static component of the model compensation error representing the virtual control thrust control law. The estimated value, the model compensation error is caused by uncertain nonlinearity and physical parameter estimation error; and These represent the first nonlinear gain matrix and the first nonlinear robust feedback gain, respectively. and These can respectively ensure the stability of the designed controller and enable Meets robust performance requirements; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between them, when When it approaches zero, It becomes very small or converges to zero. This represents the joint angular acceleration of a multi-joint hydraulic robotic arm. and transition joint angular acceleration Conversion error between; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between Projection mapping, Denotes the first projection mapping function. This represents the preset first normal value matrix; High-frequency components representing the model compensation error of the virtual control thrust control law; This represents the second linear regression matrix formed by the parameter estimation residuals; Represents the estimated value of the first dynamic parameter matrix and the first dynamic parameter matrix The error between; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angular velocity Tracking error between This indicates the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angle Tracking error between; The positive gain matrix ensures that the derivative of the Lyapunov control function of the designed nonlinear system is less than or equal to zero, thus maintaining the stability of the entire nonlinear robust controller. This represents the desired joint angular velocity of a multi-joint hydraulic robotic arm; This represents the residual term in the friction estimate; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust The error between them , and These represent the oil pressure in the inlet and outlet chambers of the hydraulic cylinders for each joint, respectively. and Let represent the desired internal friction state and its derivative, respectively.

[0040] The first robustness constraint is as follows:

[0041]

[0042] in, Static components representing model compensation error The estimation error, This represents an arbitrarily small preset first error parameter. .

[0043] The reference trajectory of the multi-joint hydraulic manipulator, which is input to the virtual control thrust control law, includes the desired joint angles of the multi-joint hydraulic manipulator. and desired joint angular velocity And the oil pressure in the oil inlet chamber of each joint hydraulic cylinder and return oil chamber oil pressure The estimated value of the dynamic parameter matrix is ​​the estimated value of the first dynamic parameter matrix. .

[0044] The desired control flow control law based on the second robustness constraint is as follows:

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051] in, This represents the desired control flow rate for a multi-joint hydraulic robotic arm, taking into account... Cannot be directly controlled, design make Approaching zero As a nonlinear robust control input for the pressure dynamics of the hydraulic manipulator cavity; and This represents the second adaptive model compensation control law and the second robust feedback control law. and These represent the second dynamic compensation control law and the second fast dynamic error compensation control law, respectively. and These represent the second linear feedback control law and the second nonlinear robust feedback control law, respectively. , and These represent the second dynamic parameter matrix respectively. The sixth dynamic parameter Seventh dynamic parameter and the eighth dynamic parameter The estimated value; and These represent the virtual control thrust of the multi-joint hydraulic robotic arm. derivative The incalculable and computable parts, This is caused by parameter estimation errors and uncertain nonlinearities, which will be suppressed through robust feedback terms. It will be added to the model compensation; This represents the dynamic compensation term caused by backstepping; The static component representing the model compensation error of the desired flow control law. The estimated value; and These represent the second nonlinear gain matrix and the second nonlinear robust feedback gain, respectively. To ensure the stability of the designed controller, Make It can meet the robust performance requirements; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust The error between them This represents the equivalent thrust of each joint hydraulic cylinder in a jointed hydraulic robotic arm. The derivative and virtual control thrust The error between the derivatives; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust Error between Projection mapping, Denotes the first projection mapping function. This represents the preset second normal value matrix; and These represent the weighting coefficients of the first and second error dynamics in the Lyapunov equations, respectively. This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between; and These represent the static and high-frequency components of the model compensation error for the desired flow control law, respectively. , and These represent the second dynamic parameter matrix respectively. The sixth dynamic parameter and its estimated value Seventh dynamic parameter and its estimated value and the eighth dynamic parameter and its estimated value The error between; This indicates the joint angles of a multi-joint hydraulic robotic arm. and These represent the estimated values ​​of the joint angular acceleration of the multi-joint hydraulic robotic arm and their errors relative to the original values, respectively. The static component of the model compensation error representing the virtual control thrust control law. The estimated value, This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between Projection mapping; and Representing the dynamic parameter matrix respectively The estimated value and its derivative, the dynamic parameter matrix Including the first dynamic parameter matrix Second dynamic parameter matrix .

[0052]

[0053] in, Static components representing model compensation error The estimation error, This represents an arbitrarily small preset second error parameter. .

[0054] The valve orifice flow mapping function is as follows:

[0055]

[0056] in, This indicates the control voltage of the valve in a multi-joint hydraulic robotic arm; and These represent the flow-voltage amplification factors of the first and second valve ports, respectively. and These represent the first and second nonlinear mapping functions of pressure to valve orifice flow rate versus voltage, respectively. This represents the desired control flow rate for a multi-joint hydraulic robotic arm.

[0057] The online adaptive law for the parameters is as follows:

[0058]

[0059]

[0060]

[0061] ,

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072] ,

[0073] ,

[0074] ,

[0075] ,

[0076] in, Represents the dynamic parameter matrix The update rate of the estimated values, and These represent the estimated values ​​of the first dynamic parameter matrix. The estimated values ​​of the second dynamic parameter matrix The update rate; Represents a saturation function; Indicates the second projection mapping function; This represents the i-th adaptive gain coefficient matrix; This represents the i-th adaptive function; This represents the i-th adaptive linear regression matrix after filtering; Indicates the prediction error. and These represent the first and second prediction errors, respectively. This represents the first adaptive linear regression matrix. and These represent the second adaptive linear regression matrix after filtering and its transformed matrix, respectively. and These represent the first dynamic parameter matrix after filtering. Second dynamic parameter matrix The regression equation is constructed using generalized momentum. The regression equation can avoid dealing with unmeasurable angular acceleration. and pressure change rate and Dependence, This represents the filtered value of the input to the first adaptive equation; The derivative of the filtered generalized momentum of the robotic arm; Represents the second dynamic parameter matrix The eighth dynamic parameter in; The derivative of the equivalent thrust of each joint hydraulic cylinder after filtering; Represents a zero matrix; and Let represent the first generalized momentum and its derivative, respectively. and Let represent the second generalized momentum and its derivative, respectively; The generalized momentum of the robotic arm; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. Indicates the preset time value; and These represent the inputs to the first and second adaptive equations, respectively. and These represent the first and second adaptive linear regression matrices, respectively. and Let represent the desired internal friction state and its derivative, respectively; and These represent the differentiating filter and the low-pass filter, respectively. express or Input volume, This represents the adaptive function after filtering.

[0077] A parameter adaptive law based on generalized momentum is constructed to compensate for the model parameter uncertainties in the overall dynamic model and dynamic friction improvement model of the hydraulic manipulator.

[0078] The adaptive robust control method based on dynamic friction precision compensation proposed in this invention can achieve precise compensation control of nonlinear friction force during low-speed processes in the movement of multi-joint hydraulic robotic arms, and ensure the stability of the control system, reduce the motion tracking error of the end effector of the multi-joint hydraulic robotic arm, and improve the control accuracy of the end effector, thereby enhancing the workability of the robotic arm in more severe working environments.

[0079] The beneficial effects of this invention are:

[0080] 1. This invention achieves real-time estimation of low-speed dynamic friction force in multi-joint hydraulic robotic arms by establishing an improved dynamic friction model and designing a method for obtaining the nominal value of internal friction state. This solves the problem of the difficulty in describing the nonlinear dynamics of friction force in actual low-speed robotic arm operations.

[0081] 2. This invention proposes a nonlinear motion control method for hydraulic robotic arms based on dynamic friction precision compensation, which reduces the tracking error at the end of the robotic arm and improves control performance while ensuring the overall stability of the control system. Attached Figure Description

[0082] Figure 1 This is a block diagram of the nonlinear motion control system for a hydraulic robotic arm based on precise dynamic friction compensation designed in this invention.

[0083] Figure 2 These are angle reference trajectory diagrams in joint space used in the two groups during the experimental verification of this invention. Figure 2 (a) is the angle reference trajectory diagram in joint space used in group 1 of the experimental verification of this invention. Figure 2 (b) is the angle reference trajectory diagram in the joint space used in group 2 of the experimental verification of this invention;

[0084] Figure 3This is the expected end-effector path and its reference trajectory diagram for each joint used in group 3 of the experimental verification of this invention, wherein... Figure 3 (a) is the terminal expected path diagram used in group 3 in the experimental verification of this invention. Figure 3 (b) is a reference trajectory diagram of each joint used in group 3 in the experimental verification of this invention;

[0085] Figure 4 This is a comparison chart of the angular velocity reference trajectories and tracking errors used in Group 1 and Group 2 during the experimental verification of this invention. Figure 4 (a) is the expected angular velocity reference trajectory diagram used in group 1 of the experimental verification of this invention. Figure 4 (b) is the expected angular velocity reference trajectory diagram used in group 2 of the experimental verification of this invention. Figure 4 (c) is a schematic diagram of the tracking error used in group 1 during the experimental verification of this invention. Figure 4 (d) is a schematic diagram of the tracking error used in group 2 during the experimental verification of this invention;

[0086] Figure 5 This is a schematic diagram of the tracking error in each dimension of Cartesian space for group 3 in the experimental verification of this invention, wherein, Figure 5 (a) is a schematic diagram of the tracking error in the X-axis dimension in Cartesian space for group 3 in the experimental verification of this invention. Figure 5 (b) is a schematic diagram of the tracking error in the Y-axis dimension in Cartesian space for group 3 in the experimental verification of this invention. Figure 5 (c) is a schematic diagram of the tracking error in the Z-axis dimension in Cartesian space in group 3 during the experimental verification of this invention;

[0087] Figure 6 This invention presents a diagram showing the internal friction state and corresponding friction compensation terms obtained using different methods, wherein... Figure 6 (a) is an internal friction state diagram obtained based on the method of the present invention. Figure 6 (b) is a diagram of friction compensation terms obtained based on the method of the present invention. Figure 6 (c) is the internal friction state diagram obtained based on the internal friction state observer. Figure 6 (d) is the friction compensation term diagram obtained by the internal friction state observer. Figure 6 Figure 7(e) is the internal friction state diagram obtained without a switching factor, and Figure 7(f) is the friction compensation term diagram obtained without a switching factor. Detailed Implementation

[0088] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. The specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0089] like Figure 1 As shown, the nonlinear motion control method for hydraulic robotic arms based on dynamic friction precision compensation of the present invention is as follows:

[0090] Step 1: Taking into account the high-order, multi-degree-of-freedom coupling and nonlinear friction characteristics of the hydraulic manipulator, establish a system dynamic parameterization and dynamic friction improvement model of the multi-joint hydraulic manipulator under the constraints of the dynamic model.

[0091] The system dynamics parameterization and dynamic friction improvement model of the multi-joint hydraulic robotic arm are as follows:

[0092]

[0093]

[0094]

[0095]

[0096] in, , and Let these represent the inertial dynamics matrix, Coriolis force matrix, and gravity matrix of the multi-joint hydraulic manipulator, respectively. ; and These represent the joint angular velocity and joint angular acceleration of a multi-joint hydraulic robotic arm, respectively. This indicates the joint angles of a multi-joint hydraulic robotic arm. ; Denotes a non-singular joint Jacobian matrix. ; and Let represent the equivalent thrust and its derivative of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm. ; Represents frictional torque. , , This represents the first linear regression matrix. Denotes the first dynamic parameter matrix. , , , , and These represent the first, second, third, fourth, and fifth dynamic parameters, respectively, which represent stiffness, damping coefficient, Coulomb friction coefficient, viscous friction coefficient, and the bounded deviation of the first lumped dynamics. Indicates the total modeling error of the first set. The bounded deviation value, the first lumped modeling error Including external disturbances and other terms that are difficult to model, ; , and These represent the sixth, seventh, and eighth dynamic parameters, respectively. , This represents the second dynamic parameter matrix. and These represent the bounded deviations in flow rate for rodless cavity dynamics and rod cavity dynamics, respectively. , Indicates the effective bulk modulus of hydraulic oil; , and These represent the first, second, and third inherent hardware parameters, respectively. , , , and These represent the contact areas of the oil inlet and return chambers of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and These represent the total compressible volumes of the oil inlet and return chambers of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm, respectively. This indicates the control flow rate of the valve in a multi-joint hydraulic robotic arm. , and These represent the actual flow rates of the inlet and outlet oil chambers of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm, respectively. Indicates the total modeling error of the second set. The bounded deviation value, , and Let represent the bounded deviation of flow rate in the rodless cavity dynamics and the rod cavity dynamics of the multi-joint hydraulic manipulator, respectively. ; Let represent the parameter set of the j-th dynamic parameter at each joint of a multi-joint hydraulic robotic arm. and These represent the dynamic parameters on the first joint and its j-th element, respectively. and Let i and j represent the dynamic parameters of the i-th joint and their j-th elements, respectively. and These represent the dynamic parameters of the nth joint and its j-th element, respectively. and These represent the dynamic parameters on the i-th joint. Stiffness and damping coefficient in and These represent the sets of stiffness and damping coefficients for each joint. ; and These represent the dynamic parameters on the i-th joint. The Coulomb coefficient of friction and the viscous coefficient of friction in the figure. and Let these represent the sets of Coulomb friction coefficients and viscous friction coefficients for each joint. ; , and These represent the dynamic parameters on the i-th joint. The first ensemble modeling error The nominal value and the bounded deviation of the flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The nominal value; This represents the switching factor; when the angular velocity exceeds a critical value proportional to the sampling rate, Can be avoided Instability in digital implementation; and Let them represent the internal friction state and its derivative, respectively. , Represents a diagonal matrix. These represent the internal friction states of the 1st, 2nd, ..., nth joints, respectively. This represents the decay function that decays in fractional form. This represents the damping coefficient related to velocity; Represents the identity matrix; Represents the differentiability parameter matrix, This can ensure frictional torque Differentiability; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. a diagonal matrix; This represents the positive matrix of the Stribeck effect.

[0097] First linear regression matrix Specifically as follows:

[0098]

[0099]

[0100]

[0101]

[0102]

[0103] in, Represents the angular velocity of the i-th joint. The switching factor, and These respectively indicate that it can be adjusted to control. The positive constants of steepness and angular offset, This represents the differentiability parameter matrix of the i-th joint. and These represent the upper and lower cutoff angular velocities selected based on the sampling rate implemented digitally.

[0104] Despite lumped modeling error It is unknown and changes over time, but it can be decomposed into slowly changing nominal values. and bounded deviation value ,Right now This can be further expressed as follows:

[0105]

[0106] in, , and These represent the total modeling error of the first set. The nominal value and the bounded deviation of the flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The nominal value.

[0107] The specific constraints of the dynamic model are as follows:

[0108]

[0109]

[0110]

[0111]

[0112] in, The inertial dynamics matrix of a multi-joint hydraulic robotic arm The derivative, and These represent the inertial dynamics matrices of the multi-joint hydraulic robotic arm. The minimum and maximum eigenvalues; This indicates the driving torque of each joint in a multi-joint hydraulic robotic arm. ; and These represent the oil pressure in the inlet and outlet chambers of each hydraulic cylinder. ; This represents the matrix of the i-th dynamic parameters. Denotes the physically feasible set of the i-th dynamic parameter matrix. and Represent the i-th dynamic parameter matrix respectively The minimum and maximum values; , and These represent the total modeling error of the first set. bounded deviation value Bounded deviation of flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The threshold; , , , and All of them are known scalars.

[0113] Inertia matrix It is a positive definite symmetric matrix, a dynamic matrix. It is obliquely symmetric, and the boundaries of parameter uncertainty and uncertainty nonlinearity are known.

[0114] Step 2: Based on the dynamic friction dynamics characteristics, and according to the system dynamics parameterization and the improved dynamic friction model, an improved LuGre friction model compensation term is designed using the expected compensation method. The improved LuGre friction model compensation term can obtain the nominal value of the internal friction state, realizing real-time updating of the unmeasurable internal friction state and real-time estimation of the friction compensation term.

[0115] The improved LuGre friction model compensation terms are as follows:

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122] in, Represents frictional torque The estimated value is the friction compensation term. This represents the expected first linear regression matrix. This represents the estimated value of the first dynamic parameter matrix; and Representing the desired internal friction state and its derivative, respectively, can be expressed as: and ; Represents the zero vector; This represents an estimate of the frictional torque. and frictional torque The error between them Represents the estimated value of the first dynamic parameter matrix and the first dynamic parameter matrix The error between; This represents the residual term in the friction estimate; and These represent the internal friction states respectively. and its derivative The explicit expression, i.e., the internal friction state and its derivative The solution, and , and These represent the first and second parameters within the preset feasible set of angular velocities, respectively. This represents the desired joint angular velocity of a multi-joint hydraulic robotic arm. a diagonal matrix; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angular velocity Tracking error between A diagonal matrix.

[0123] The improved LuGre friction model uses the idea of ​​expected compensation in its compensation terms, based on the derivative of the expected internal friction state. Continuously update the internal friction state .

[0124] The reference trajectory of the multi-joint hydraulic manipulator, with the input of the improved LuGre friction model compensation term, is the desired joint angular velocity of the multi-joint hydraulic manipulator. The estimated value of the dynamic parameter matrix of the multi-joint hydraulic manipulator is the estimated value of the first dynamic parameter matrix. .

[0125] Step 3: Design an adaptive robust controller for a multi-joint hydraulic manipulator based on nonlinear dynamic friction compensation. Input the reference trajectory of the multi-joint hydraulic manipulator into the adaptive robust controller and the improved LuGre friction model compensation term. The adaptive robust controller outputs the estimated value of the dynamic parameter matrix of the multi-joint hydraulic manipulator into the improved LuGre friction model compensation term. The improved LuGre friction model compensation term outputs the friction force compensation term into the adaptive robust controller. The adaptive robust controller outputs the control voltage of the valve of the multi-joint hydraulic manipulator to control the multi-joint hydraulic manipulator. Input the state feedback quantity of the multi-joint hydraulic manipulator during motion into the improved LuGre friction model compensation term and the adaptive robust controller of the multi-joint hydraulic manipulator to complete the closed loop and realize the nonlinear motion control of the multi-joint hydraulic manipulator.

[0126] The adaptive robust controller for the multi-joint hydraulic manipulator includes a virtual control thrust control law based on a first robustness constraint, a desired control flow control law based on a second robustness constraint, a valve orifice flow mapping function, and an online parameter adaptive law. The reference trajectory of the multi-joint hydraulic manipulator is input into the virtual control thrust control law. The online parameter adaptive law inputs the estimated values ​​of the dynamically updated dynamic parameter matrix into the improved LuGre friction model compensation term, the virtual control thrust control law, and the desired control flow control law, respectively. The improved LuGre friction model compensation term outputs a friction compensation term into the virtual control thrust control law. The virtual control thrust control law outputs the virtual control thrust into the desired control flow control law. The desired control flow control law outputs the desired control flow into the valve orifice flow mapping function. The valve orifice flow mapping function outputs the control voltage of the valve of the multi-joint hydraulic manipulator to control the multi-joint hydraulic manipulator.

[0127] The virtual control thrust law based on the first robustness constraint is as follows:

[0128]

[0129]

[0130]

[0131]

[0132]

[0133]

[0134] in, This represents the virtual control thrust of a multi-joint hydraulic robotic arm. and These represent the first adaptive model compensation control law and the first robust feedback control law, respectively. and These represent the first dynamic compensation control law and the first fast dynamic error compensation control law, respectively. and These represent the first linear feedback control law and the first nonlinear robust feedback control law, respectively. and These represent the transition joint angular velocity and transition joint angular acceleration of the multi-joint hydraulic robotic arm, respectively. This indicates the friction compensation term; Represents the first dynamic parameter matrix The fifth dynamic parameter in The estimated value; The static component of the model compensation error representing the virtual control thrust control law. The estimated value, the model compensation error is caused by uncertain nonlinearity and physical parameter estimation error; and These represent the first nonlinear gain matrix and the first nonlinear robust feedback gain, respectively. and These can respectively ensure the stability of the designed controller and enable Meets robust performance requirements; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between them, when When it approaches zero, It becomes very small or converges to zero. This represents the joint angular acceleration of a multi-joint hydraulic robotic arm. and transition joint angular acceleration Conversion error between; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between Projection mapping, Denotes the first projection mapping function. This represents the preset first normal value matrix; High-frequency components representing the model compensation error of the virtual control thrust control law; This represents the second linear regression matrix formed by the parameter estimation residuals; Represents the estimated value of the first dynamic parameter matrix and the first dynamic parameter matrix The error between; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angular velocity Tracking error between This indicates the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angle Tracking error between; The positive gain matrix ensures that the derivative of the Lyapunov control function of the designed nonlinear system is less than or equal to zero, thus maintaining the stability of the entire nonlinear robust controller. This represents the desired joint angular velocity of a multi-joint hydraulic robotic arm; This represents the residual term in the friction estimate; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust The error between them , and These represent the oil pressure in the inlet and outlet chambers of the hydraulic cylinders for each joint, respectively. and Let represent the desired internal friction state and its derivative, respectively.

[0135] The first robustness constraint is as follows:

[0136]

[0137] in, Static components representing model compensation error The estimation error, This represents an arbitrarily small preset first error parameter. .

[0138] The reference trajectory of the multi-joint hydraulic manipulator, which is input to the virtual control thrust control law, includes the desired joint angles of the multi-joint hydraulic manipulator. and desired joint angular velocity And the oil pressure in the oil inlet chamber of each joint hydraulic cylinder and return oil chamber oil pressure The estimated value of the dynamic parameter matrix is ​​the estimated value of the first dynamic parameter matrix. .

[0139] The desired control flow control law based on the second robustness constraint is as follows:

[0140]

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] in, This represents the desired control flow rate for a multi-joint hydraulic robotic arm, taking into account... Cannot be directly controlled, design make Approaching zero As a nonlinear robust control input for the pressure dynamics of the hydraulic manipulator cavity; and This represents the second adaptive model compensation control law and the second robust feedback control law. and These represent the second dynamic compensation control law and the second fast dynamic error compensation control law, respectively. and These represent the second linear feedback control law and the second nonlinear robust feedback control law, respectively. , and These represent the second dynamic parameter matrix respectively. The sixth dynamic parameter Seventh dynamic parameter and the eighth dynamic parameter The estimated value; and These represent the virtual control thrust of the multi-joint hydraulic robotic arm. derivative The incalculable and computable parts, This is caused by parameter estimation errors and uncertain nonlinearities, which will be suppressed through robust feedback terms. It will be added to the model compensation; This represents the dynamic compensation term caused by backstepping; The static component representing the model compensation error of the desired flow control law. The estimated value; and These represent the second nonlinear gain matrix and the second nonlinear robust feedback gain, respectively. To ensure the stability of the designed controller, Make It can meet the robust performance requirements; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust The error between them This represents the equivalent thrust of each joint hydraulic cylinder in a jointed hydraulic robotic arm. The derivative and virtual control thrust The error between the derivatives; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust Error between Projection mapping, Denotes the first projection mapping function. This represents the preset second normal value matrix; and These represent the weighting coefficients of the first and second error dynamics in the Lyapunov equations, respectively. This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between; and These represent the static and high-frequency components of the model compensation error for the desired flow control law, respectively. , and These represent the second dynamic parameter matrix respectively. The sixth dynamic parameter and its estimated value Seventh dynamic parameter and its estimated value and the eighth dynamic parameter and its estimated value The error between; This indicates the joint angles of a multi-joint hydraulic robotic arm. and These represent the estimated values ​​of the joint angular acceleration of the multi-joint hydraulic robotic arm and their errors relative to the original values, respectively. The static component of the model compensation error representing the virtual control thrust control law. The estimated value, This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between Projection mapping; and Representing the dynamic parameter matrix respectively The estimated value and its derivative, the dynamic parameter matrix Including the first dynamic parameter matrix Second dynamic parameter matrix .

[0147]

[0148] in, Static components representing model compensation error The estimation error, This represents an arbitrarily small preset second error parameter. .

[0149] First projection mapping function Specifically as follows:

[0150]

[0151] in, , and These represent the estimated values ​​and upper and lower limits of the static components of the model compensation error, respectively.

[0152] The valve orifice flow mapping function is as follows:

[0153]

[0154] in, This indicates the control voltage of the valve in a multi-joint hydraulic robotic arm; and These represent the flow-voltage amplification factors of the first and second valve ports, respectively. and These represent the first and second nonlinear mapping functions of pressure to valve orifice flow rate versus voltage, respectively. This represents the desired control flow rate for a multi-joint hydraulic robotic arm.

[0155] The online adaptive law for parameters is as follows:

[0156]

[0157]

[0158]

[0159] ,

[0160]

[0161]

[0162]

[0163]

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170] ,

[0171] ,

[0172] ,

[0173] ,

[0174] in, Represents the dynamic parameter matrix The update rate of the estimated values, and These represent the estimated values ​​of the first dynamic parameter matrix. The estimated values ​​of the second dynamic parameter matrix The update rate; Represents a saturation function; Indicates the second projection mapping function; This represents the i-th adaptive gain coefficient matrix; This represents the i-th adaptive function; This represents the i-th adaptive linear regression matrix after filtering; Indicates the prediction error. and These represent the first and second prediction errors, respectively. This represents the first adaptive linear regression matrix. and These represent the second adaptive linear regression matrix after filtering and its transformed matrix, respectively. and These represent the first dynamic parameter matrix after filtering. Second dynamic parameter matrix The regression equation is constructed using generalized momentum. The regression equation can avoid dealing with unmeasurable angular acceleration. and pressure change rate and Dependence, This represents the filtered value of the input to the first adaptive equation; The derivative of the filtered generalized momentum of the robotic arm; Represents the second dynamic parameter matrix The eighth dynamic parameter in; The derivative of the equivalent thrust of each joint hydraulic cylinder after filtering; Represents a zero matrix; and Let represent the first generalized momentum and its derivative, respectively. and Let represent the second generalized momentum and its derivative, respectively; The generalized momentum of the robotic arm; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. Indicates the preset time value; and These represent the inputs to the first and second adaptive equations, respectively. and These represent the first and second adaptive linear regression matrices, respectively. and Let represent the desired internal friction state and its derivative, respectively; and These represent the differentiating filter and the low-pass filter, respectively. express or Input volume, This represents the adaptive function after filtering.

[0175] A parameter adaptive law based on generalized momentum is constructed to compensate for the model parameter uncertainties in the overall dynamic model and dynamic friction improvement model of the hydraulic manipulator.

[0176] Saturation function Specifically as follows:

[0177]

[0178] in, This indicates the preset rate limit.

[0179] Projection mapping Specifically as follows:

[0180]

[0181] in, Represents the identity matrix; This represents the feasible set of adaptive parameters. express The boundary; Indicates in The unit normal vector pointing outwards at that location.

[0182] It can be updated by the following formula:

[0183]

[0184] in, Represents the i-th forgetting factor; This represents the upper bound of the eigenvalues ​​of the preset i-th adaptive gain.

[0185] Specific embodiments of the present invention are as follows:

[0186] Experiments were conducted using a multi-degree-of-freedom hydraulic robotic arm to test the aforementioned control method. The results were compared with those of the same controller that simplifies friction to Coulomb friction and viscous friction, verifying the control effectiveness of the proposed method. The controller proposed in this invention is denoted as C1, and the controller after simplified friction compensation is denoted as C2. The only difference between C2 and C1 is the friction compensation term, which is simplified to... During verification, the parameter selections for both controllers C1 and C2 are shown in Table 1.

[0187] Table 1 Controller Parameter Selection

[0188]

[0189] Comparative experiments were conducted in joint space and Cartesian space. In the joint space experiment, to eliminate the influence of joint coupling and gravity on the friction compensation effect, a swing joint was selected to track the desired trajectory, as shown in the figure. Figure 2 (a) and Figure 2 As shown in (b), Group 1: Low-speed sine curve tracking experiment. The desired angular velocity is set to... Group 2: Low-speed point-to-point S-curve tracking experiment. The desired trajectory is an S-curve from 0.66 rad to 1.71 rad, with a maximum angular velocity of... The maximum angular acceleration is .

[0190] Furthermore, a comparative experiment in Cartesian space was set up as Group 3: Multi-degree-of-freedom trajectory tracking experiment. The desired trajectory of the end effector. Set as a third-order differentiable trajectory, such as Figure 3 As shown in (a), it resembles the symbol "∞". Considering the existence of redundant degrees of freedom, the end effector of the hydraulic manipulator is set to be perpendicular to the horizontal plane. The manipulator's posture information is converted into the desired angles of the four joints through inverse kinematics operations. Then, the corresponding desired angular velocity trajectory is tracked by controllers C1 and C2 respectively. like Figure 3 As shown in (b).

[0191] To compare the controller's performance, the following metrics were selected:

[0192] , ,

[0193] in, and These represent the first and second performance indicators, respectively. Represents the L2 norm; This represents the tracking error at time t; Indicates the total time period. This represents the performance index of normalized error. Indicates the speed of the joint or end effector.

[0194] The comparative experimental results of group 1 and group 2 are as follows: Figure 4 As shown. Figure 4 (a) and Figure 4 As shown in (b), the angular velocity reference trajectories used in Group 1 and Group 2 are as follows: Figure 4 (c) and Figure 4 As shown in (d), the corresponding controller tracking error is represented. Clearly, the larger error only occurs in the lower angular velocity phase, confirming that the nonlinear characteristics of low-speed friction are the main factor limiting the system's control accuracy. In Group 1, the swing joint of the hydraulic manipulator frequently requires low-speed movement. During this period, the switching factor of the C1 controller is close to 1, and dynamic friction precision compensation takes effect, effectively reducing the peak error compared to the C2 controller. In Group 2, as a typical point-to-point motion, the system experiences frequent start-stop movements, i.e., the speed increases from zero and then gradually decreases back to zero. The compensation control during the acceleration phase is similar to that in Group 1. During deceleration, the model exhibits friction hysteresis, indicating the effectiveness of estimating the internal friction state. Figure 4 As shown in the magnified images of the two parts in (d), the effective compensation during the start-stop phase is consistent in both the forward and reverse movements of the hydraulic joint. Meanwhile, experimental results indicate that converting the accurate dynamic friction model into a combination of Coulomb friction and viscous friction in the high-speed phase by introducing a switching factor is a reasonable approach.

[0195] The results of the comparative experiment for group 3 are as follows: Figure 5 of (a) Figure 5 (b) and Figure 5 As shown in (c), the boxed portion represents the period when the joint is in low-speed motion, corresponding to... Figure 3 The desired angular velocity trajectory is shown in (b). Clearly, low-speed joint movement is quite common. Experimental results demonstrate that the C1 controller has an advantage over the C2 controller in tracking performance across all dimensions, particularly along the Y-axis.

[0196] The quantitative analysis results of the control performance of the three groups of experiments are shown in Table 2. The results show that the C1 controller has a significant advantage over the C2 controller.

[0197] Table 2 Quantitative Analysis of Control Performance

[0198]

[0199] The internal friction states and corresponding friction compensation terms obtained by different methods in the comparative experiment are as follows: Figure 6 As shown in the figure. The data in the figure comes from the swing joint of the robotic arm in experimental group 3. The data was obtained using the method proposed in this invention. and like Figure 6 (a) and Figure 6 As shown in (b), This represents the expected angular velocity of the first joint. and These represent the upper and lower bounds of the velocity switching in the nonlinear friction model, which can avoid unnecessary oscillations and provide effective friction compensation as much as possible. Figure 6 (c) and Figure 6 The data in (d) comes from the method of the internal friction state observer, which, compared to the method proposed in this invention, has advantages in... The observations are highly sensitive to measurement noise and phase hysteresis caused by filters, and exhibit fluctuations at low speeds (as shown in the magnified inset). These fluctuations lead to... The rapid and significant changes caused the design's dynamic compensation control law to be affected. Anomalies can occur, thereby suppressing system bandwidth, limiting feedback gain, and potentially causing system instability. For example... Figure 6 (e) and Figure 6 As shown in (f), this illustrates the situation without a switching factor. The internal friction force observation results at that time, once the speed exceeds the threshold (as shown in the darker color part in the figure). Significant oscillations will occur. Therefore, the friction model proposed in this invention avoids oscillations and enables the controller to achieve better friction compensation.

[0200] Compared to controllers based on traditional friction model compensation, this invention significantly reduces joint tracking error and shortens transient response time during low-speed motion. This demonstrates that the nonlinear motion control method for hydraulic robotic arms based on dynamic friction precision compensation, designed in this invention, exhibits superior transient response performance and better robustness. It effectively compensates for the highly nonlinear frictional force under low-speed motion conditions, reducing tracking error at the robotic arm's end effector while ensuring control system stability and improving control performance. Compared to other friction models and internal friction observation methods, the friction model and internal friction estimation method proposed in this invention effectively avoid result oscillations and provide more stable friction compensation to the controller.

[0201] The above content is merely a technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solution based on the technical concept proposed in this invention shall fall within the scope of protection of the claims of this invention.

Claims

1. A hydraulic manipulator nonlinear motion control method based on dynamic friction accurate compensation, characterized in that, include: Step 1: Establish a system dynamic parameterization and dynamic friction improvement model for the multi-joint hydraulic manipulator under dynamic model constraints; Step 2: Based on the system dynamics parameterization and dynamic friction improvement model, design the improved LuGre friction model compensation term using the expected compensation method; Step 3: Design an adaptive robust controller for the multi-joint hydraulic manipulator. Input the reference trajectory of the multi-joint hydraulic manipulator into the adaptive robust controller and the improved LuGre friction model compensation term. The adaptive robust controller outputs the estimated value of the dynamic parameter matrix of the multi-joint hydraulic manipulator to the improved LuGre friction model compensation term. The improved LuGre friction model compensation term outputs the friction compensation term to the adaptive robust controller. The adaptive robust controller outputs the control voltage of the valve of the multi-joint hydraulic manipulator to control the multi-joint hydraulic manipulator. Input the state feedback quantity of the multi-joint hydraulic manipulator during movement into the improved LuGre friction model compensation term and the adaptive robust controller of the multi-joint hydraulic manipulator to complete the closed loop and realize the nonlinear motion control of the multi-joint hydraulic manipulator. In the first step, the system dynamics parameterization and dynamic friction improvement model of the multi-joint hydraulic manipulator are as follows: in, , and These represent the inertial dynamics matrix, Coriolis force matrix, and gravity matrix of the multi-joint hydraulic manipulator, respectively. and These represent the joint angular velocity and joint angular acceleration of a multi-joint hydraulic robotic arm, respectively. Represents a non-singular joint Jacobian matrix; and These represent the equivalent thrust and its derivative of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm; Represents frictional torque. , This represents the first linear regression matrix. Denotes the first dynamic parameter matrix. , , , , and These represent the first, second, third, fourth, and fifth dynamic parameters, respectively. Indicates the total modeling error of the first set. The bounded deviation value; , and These represent the sixth, seventh, and eighth dynamic parameters, respectively. , Represents the second dynamic parameter matrix; , and These represent the first, second, and third inherent hardware parameters, respectively. , , , and These represent the contact areas of the oil inlet and return chambers of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and These represent the total compressible volumes of the oil inlet and return chambers of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm, respectively. This indicates the control flow rate of the valve in a multi-joint hydraulic robotic arm. , and These represent the actual flow rates of the inlet and outlet oil chambers of the hydraulic cylinders at each joint of the multi-joint hydraulic robotic arm, respectively. Indicates the total modeling error of the second set. The bounded deviation value, , and These represent the bounded deviations in flow rate for the rodless and rod-type chamber dynamics of a multi-joint hydraulic manipulator, respectively. Let represent the parameter set of the j-th dynamic parameter at each joint of a multi-joint hydraulic robotic arm. and These represent the dynamic parameters on the first joint and its j-th element, respectively. and Let i and j represent the dynamic parameters of the i-th joint and their j-th elements, respectively. and These represent the dynamic parameters of the nth joint and its j-th element, respectively. and These represent the dynamic parameters on the i-th joint. Stiffness and damping coefficient in and These represent the sets of stiffness and damping coefficients for each joint; and These represent the dynamic parameters on the i-th joint. The Coulomb coefficient of friction and the viscous coefficient of friction in the figure. and These represent the sets of Coulomb friction coefficients and viscous friction coefficients for each joint, respectively. , and These represent the dynamic parameters on the i-th joint. The first ensemble modeling error The nominal value and the bounded deviation of the flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The nominal value; Indicates the switching factor; and Let them represent the internal friction state and its derivative, respectively. , Represents a diagonal matrix. These represent the internal friction states of the 1st, 2nd, ..., nth joints, respectively. Represents the decay function; Represents the identity matrix; Represents the differentiability parameter matrix; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. a diagonal matrix; This represents the positive matrix of the Stribeck effect.

2. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 1, characterized in that: In the first step described above, the constraints of the dynamic model are as follows: in, The inertial dynamics matrix of a multi-joint hydraulic robotic arm The derivative, and These represent the inertial dynamics matrices of the multi-joint hydraulic robotic arm. The minimum and maximum eigenvalues; This indicates the driving torque of each joint in a multi-joint hydraulic robotic arm. and These represent the oil pressure in the inlet and outlet chambers of the hydraulic cylinders for each joint, respectively. This represents the matrix of the i-th dynamic parameters. Denotes the physically feasible set of the i-th dynamic parameter matrix. and Represent the i-th dynamic parameter matrix respectively The minimum and maximum values; , and These represent the total modeling error of the first set. bounded deviation value Bounded deviation of flow rate in rodless cavity dynamics Bounded deviation of flow rate from rod cavity dynamics The threshold.

3. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 1, characterized in that: In the second step, the compensation term of the improved LuGre friction model is as follows: in, Represents frictional torque The estimated value is the friction compensation term. This represents the expected first linear regression matrix. This represents the estimated value of the first dynamic parameter matrix; and Let represent the desired internal friction state and its derivative, respectively; Represents the zero vector; This represents an estimate of the frictional torque. and frictional torque The error between them Represents the estimated value of the first dynamic parameter matrix and the first dynamic parameter matrix The error between; This represents the residual term in the friction estimate; and These represent the internal friction states respectively. and its derivative Explicit expression. and These represent the first and second parameters within the preset feasible set of angular velocities, respectively. This represents the desired joint angular velocity of a multi-joint hydraulic robotic arm. a diagonal matrix; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angular velocity Tracking error between a diagonal matrix; The improved LuGre friction model compensation term is based on the derivative of the desired internal friction state. Continuously update the internal friction state ; The reference trajectory of the multi-joint hydraulic manipulator, with the input of the improved LuGre friction model compensation term, is the desired joint angular velocity of the multi-joint hydraulic manipulator. The estimated value of the dynamic parameter matrix of the multi-joint hydraulic manipulator is the estimated value of the first dynamic parameter matrix. .

4. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 1, characterized in that: In the third step, the adaptive robust controller of the multi-joint hydraulic manipulator includes a virtual control thrust control law based on a first robustness constraint, a desired control flow control law based on a second robustness constraint, a valve orifice flow mapping function, and an online parameter adaptive law. The reference trajectory of the multi-joint hydraulic manipulator is input into the virtual control thrust control law. The online parameter adaptive law inputs the estimated values ​​of the dynamically updated dynamic parameter matrix into the improved LuGre friction model compensation term, the virtual control thrust control law, and the desired control flow control law, respectively. The improved LuGre friction model compensation term outputs a friction compensation term into the virtual control thrust control law. The virtual control thrust control law outputs virtual control thrust into the desired control flow control law. The desired control flow control law outputs the desired control flow into the valve orifice flow mapping function. The valve orifice flow mapping function outputs the control voltage of the valve of the multi-joint hydraulic manipulator to control the multi-joint hydraulic manipulator.

5. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 4, characterized in that: The virtual control thrust control law based on the first robustness constraint is as follows: in, This represents the virtual control thrust of a multi-joint hydraulic robotic arm. and These represent the first adaptive model compensation control law and the first robust feedback control law, respectively. and These represent the first dynamic compensation control law and the first fast dynamic error compensation control law, respectively. and These represent the first linear feedback control law and the first nonlinear robust feedback control law, respectively. and These represent the transition joint angular velocity and transition joint angular acceleration of the multi-joint hydraulic robotic arm, respectively. This indicates the friction compensation term; Represents the first dynamic parameter matrix The fifth dynamic parameter in The estimated value; The static component of the model compensation error representing the virtual control thrust control law. The estimated value; and These represent the first nonlinear gain matrix and the first nonlinear robust feedback gain, respectively. This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between This represents the joint angular acceleration of a multi-joint hydraulic robotic arm. and transition joint angular acceleration Conversion error between; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between Projection mapping, Denotes the first projection mapping function. This represents the preset first normal value matrix; High-frequency components representing the model compensation error of the virtual control thrust control law; This represents the second linear regression matrix; Represents the estimated value of the first dynamic parameter matrix and the first dynamic parameter matrix The error between; This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angular velocity Tracking error between This indicates the joint angular velocity of a multi-joint hydraulic robotic arm. and desired joint angle Tracking error between; Represents the positive gain matrix; This represents the desired joint angular velocity of a multi-joint hydraulic robotic arm; This represents the residual term in the friction estimate; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust The error between them , and These represent the oil pressure in the inlet and outlet chambers of the hydraulic cylinders for each joint, respectively. and Let represent the desired internal friction state and its derivative, respectively; The first robustness constraint is as follows: in, Static components representing model compensation error The estimation error, This indicates the preset first error parameter. ; The reference trajectory of the multi-joint hydraulic manipulator, which is input to the virtual control thrust control law, includes the desired joint angles of the multi-joint hydraulic manipulator. and desired joint angular velocity And the oil pressure in the oil inlet chamber of each joint hydraulic cylinder and return oil chamber oil pressure The estimated value of the dynamic parameter matrix is ​​the estimated value of the first dynamic parameter matrix. .

6. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 4, characterized in that: The desired control flow control law based on the second robustness constraint is as follows: in, This represents the desired control flow rate for a multi-joint hydraulic robotic arm. and This represents the second adaptive model compensation control law and the second robust feedback control law. and These represent the second dynamic compensation control law and the second fast dynamic error compensation control law, respectively. and These represent the second linear feedback control law and the second nonlinear robust feedback control law, respectively. , and These represent the second dynamic parameter matrix respectively. The sixth dynamic parameter Seventh dynamic parameter and the eighth dynamic parameter The estimated value; and These represent the virtual control thrust of the multi-joint hydraulic robotic arm. derivative The incalculable and computable parts; Represents the dynamic compensation term; The static component representing the model compensation error of the desired flow control law. The estimated value; and These represent the second nonlinear gain matrix and the second nonlinear robust feedback gain, respectively. This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust The error between them This represents the equivalent thrust of each joint hydraulic cylinder in a jointed hydraulic robotic arm. The derivative and virtual control thrust The error between the derivatives; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. and virtual control thrust Error between Projection mapping, Denotes the first projection mapping function. This represents the preset second normal value matrix; and These represent the weighting coefficients of the first and second error dynamics in the Lyapunov equations, respectively. This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between; and These represent the static and high-frequency components of the model compensation error for the desired flow control law, respectively. , and These represent the second dynamic parameter matrix respectively. The sixth dynamic parameter and its estimated value Seventh dynamic parameter and its estimated value and the eighth dynamic parameter and its estimated value The error between; This indicates the joint angles of a multi-joint hydraulic robotic arm. and These represent the estimated values ​​of the joint angular acceleration of the multi-joint hydraulic robotic arm and their errors relative to the original values, respectively. The static component of the model compensation error representing the virtual control thrust control law. The estimated value, This represents the joint angular velocity of a multi-joint hydraulic robotic arm. and conversion of joint angular velocity Conversion error between Projection mapping; and Representing the dynamic parameter matrix respectively The estimated value and its derivative, the dynamic parameter matrix Including the first dynamic parameter matrix Second dynamic parameter matrix ; in, Static components representing model compensation error The estimation error, This indicates a preset second error parameter. .

7. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 4, characterized in that: The valve orifice flow mapping function is as follows: in, This indicates the control voltage of the valve in a multi-joint hydraulic robotic arm; and These represent the flow-voltage amplification factors of the first and second valve ports, respectively. and These represent the first and second nonlinear mapping functions of pressure to valve orifice flow rate versus voltage, respectively. This represents the desired control flow rate for a multi-joint hydraulic robotic arm.

8. The nonlinear motion control method for a hydraulic robotic arm based on precise dynamic friction compensation according to claim 4, characterized in that: The online adaptive law for the parameters is as follows: ， in, Represents the dynamic parameter matrix The update rate of the estimated values, and These represent the estimated values ​​of the first dynamic parameter matrix. The estimated values ​​of the second dynamic parameter matrix The update rate; Represents a saturation function; Indicates the second projection mapping function; This represents the i-th adaptive gain coefficient matrix; This represents the i-th adaptive function; This represents the i-th adaptive linear regression matrix after filtering; Indicates the prediction error. and These represent the first and second prediction errors, respectively. This represents the first adaptive linear regression matrix. and These represent the second adaptive linear regression matrix after filtering and its transformed matrix, respectively. and These represent the first dynamic parameter matrix after filtering. Second dynamic parameter matrix The regression equation, This represents the filtered value of the input to the first adaptive equation; The derivative of the filtered generalized momentum of the robotic arm; Represents the second dynamic parameter matrix The eighth dynamic parameter in; The derivative of the equivalent thrust of each joint hydraulic cylinder after filtering; Represents a zero matrix; and Let represent the first generalized momentum and its derivative, respectively. and Let represent the second generalized momentum and its derivative, respectively; The generalized momentum of the robotic arm; This represents the equivalent thrust of the hydraulic cylinders at each joint of a multi-joint hydraulic robotic arm. Indicates the preset time value; and These represent the inputs to the first and second adaptive equations, respectively. and These represent the first and second adaptive linear regression matrices, respectively. and Let represent the desired internal friction state and its derivative, respectively.

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