Mechanical arm nonlinear friction identification and compensation method based on sliding mode control

Abstract

The invention provides a mechanical arm nonlinear friction identification and compensation method based on sliding mode control. The method comprises the steps that a kinetic equation of an n-degree-of-freedom mechanical arm is obtained through modeling, system friction is modeled through a Lugre friction model, a friction model equation is obtained, and model parameters of the friction model equation are estimated; measuring an output variable of the friction model equation, calculating a result torque according to the output variable, comparing the result torque with an ideal input torque, and obtaining a system friction torque loss parameter according to a comparison result; and the result is substituted into the kinetic equation of the n-degree-of-freedom mechanical arm, and the kinetic equation after nonlinear friction identification and compensation of the mechanical arm is obtained. Accurate identification and compensation of nonlinear friction of the mechanical arm are achieved, and the motion precision and stability of the mechanical arm are improved. Through combined use of a nonlinear sliding mode control method and a Lugre friction model, the influence of the nonlinear friction torque on the mechanical arm in the movement process can be effectively processed.

Description

Technical Field

[0001] This application relates to the field of nonlinear friction identification and compensation technology for robotic arms, and in particular to a method for nonlinear friction identification and compensation of robotic arms based on sliding mode control. Background Technology

[0002] Robotic arm control has broad significance, playing a crucial role in various fields and applications. In industrial production, robotic arms are widely used in automated production lines, improving production efficiency and quality. They can perform highly repetitive and precision-required tasks such as assembly, welding, handling, and painting. Robotic arms can also be used to perform tasks in hazardous or harsh environments, thus preventing human injury. This includes applications in nuclear power plants, chemical plants, deep-sea exploration, and space exploration, making robotic arm control particularly important.

[0003] Current robotic arm control algorithms still have some shortcomings. Some algorithms require significant computational resources, making them unsuitable for embedded systems or resource-constrained environments. This increases system cost and complexity. Furthermore, these algorithms may not be robust enough to external disturbances, sensor errors, or changes in the mechanical system, potentially leading to instability and degraded control performance. Some algorithms rely on predefined models and parameters, exhibiting poor adaptability to changing and unknown environments. A lack of sensing capabilities and real-time adjustment may limit the application of robotic arms in complex environments. Summary of the Invention

[0004] This application addresses the problem that certain robotic arm control algorithms require significant computational resources, making them unsuitable for embedded systems or resource-constrained environments. It provides a method for identifying and compensating for nonlinear friction in robotic arms based on sliding mode control. The method includes:

[0005] The dynamic equations of the n-degree-of-freedom manipulator are obtained by modeling, and the dynamic equations of the n-degree-of-freedom manipulator include control input vectors and friction compensation terms;

[0006] The control input vector is obtained through a nonlinear sliding mode control method, and the friction compensation term is the difference between the friction torque loss parameter and the actual measured friction torque value.

[0007] The friction of the system is modeled using the Lugre friction model to obtain the friction model equation. The friction model equation is simplified, and the parameters of the friction model equation are estimated using the nonlinear least squares method.

[0008] The output variables of the friction model equation are measured, and the resulting torque is calculated based on the output variables. The resulting torque is compared with the ideal input torque, and the system friction torque loss parameters are obtained based on the comparison results.

[0009] Substituting the system friction torque loss parameters, the actual measured friction torque value, and the control input vector into the dynamic equation of the n-degree-of-freedom manipulator, the dynamic equation of the manipulator after nonlinear friction identification and compensation is obtained.

[0010] In one feasible implementation method

[0011] The general form of the dynamic equations of the n-degree-of-freedom robotic arm is:

[0012]

[0013] In the formula, q(t) represents the position of the robotic arm, q¨(t) represents the velocity of the robotic arm, q˙(t) represents the acceleration of the robotic arm, λ represents the design constant of the robotic arm, S represents the state variable of the robotic arm, V represents the linear velocity of the robotic arm, and ω represents the angular velocity of the robotic arm.

[0014] in, The mass matrix represents the mass matrix, which is obtained based on the position of the robotic arm joints. G(q(t)) represents the Coriolis force matrix, which is obtained based on the velocity of the robotic arm joint; G(q(t)) represents the gravity vector, which is obtained based on the position of the robotic arm joint; u(t) represents the control input vector, which is the torque of the robotic arm joint motor.

[0015] The friction compensation term is represented as:

[0016]

[0017] In the formula, The friction torque loss parameter is... The difference between the actual measured friction torque values.

[0018] In one feasible implementation, the step of obtaining the control input vector through a nonlinear sliding mode control method further includes:

[0019] The control input vector, consisting of two equivalent symbols and one correction symbol, is obtained through a nonlinear sliding mode control method, and is defined as follows:

[0020]

[0021] In the formula, u(t) is the control input vector. For control gain or control matrix, For the system gain or parameter matrix, For the estimated friction compensation term, The product of the estimated mass matrix, the expected acceleration, and the measurement error. It is a nonlinear function term;

[0022] For the linear term of velocity V, N is the linear term for angular velocity ω, K is the matrix of gain coefficients, λ is the design constant, and N,K I ,R,K b This is the motor coefficient.

[0023] In one feasible implementation, the step of modeling the system friction using the Lugre friction model to obtain the friction model equations includes the following steps:

[0024] The friction model equation is obtained from the following formula:

[0025]

[0026] In the formula, F represents frictional force, and σ0z represents static frictional force. Represents kinetic friction. Represents damping force or resistance. Represents acceleration. Represents speed, Represents nonlinear terms;

[0027] in, for:

[0028]

[0029] In the formula, F c F represents the central force. S σ represents saturation force, vs represents linear velocity, and σ represents saturation force. v Represents angular velocity.

[0030] In one feasible implementation, simplifying the friction model equations and estimating the friction model equation parameters using nonlinear least squares includes the following steps:

[0031] By eliminating the internal acceleration variables in the friction model equations The friction model equations are simplified;

[0032] Assuming Estimating the parameters in the friction model equations as time-varying yields the following equation:

[0033]

[0034] The parameters are estimated using the nonlinear least squares method to obtain the final equation of the friction model. The parameters estimated by the nonlinear least squares method are expressed by the following formula:

[0035] (J T J+λI)Δθ=J T Δy;

[0036] In the formula, J represents a matrix, J T J represents the transpose matrix. T J represents the product of the transpose of J and J, called the covariance matrix of the error; λI represents the regularization term, where I is the identity matrix; Δθ represents the parameter vector to be estimated; Δy represents the measurement error.

[0037] In one feasible implementation, the resulting torque is compared with the ideal input torque, and the system friction torque loss parameter is calculated based on the comparison result using the following formula:

[0038]

[0039] In the formula, The system friction torque loss parameter, The input torque, u L (t) represents the load torque, which is the torque generated by other resistances or external forces in the system.

[0040] As described above, this application provides a method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control. The method includes: modeling an n-degree-of-freedom (DOF) robotic arm's dynamic equations, which include a control input vector and friction compensation terms; modeling the system friction using the Lugre friction model to obtain friction model equations, simplifying the friction model equations, and estimating the model parameters using nonlinear least squares; measuring the output variables of the friction model equations and calculating the resulting torque based on the output variables, comparing the resulting torque with the ideal input torque, and obtaining system friction torque loss parameters based on the comparison results; and substituting the system friction torque loss parameters, the actually measured friction torque value, and the control input vector into the dynamic equations of the n-DOF robotic arm to obtain the nonlinear friction identification and compensation dynamic equations of the robotic arm. This application, by implementing friction identification and compensation for the robotic arm, enhances the robotic arm's perception capabilities and further improves the accuracy of environmental perception and object recognition. In addition, it is necessary to improve perception fusion technology to integrate information from different sensors to provide a more comprehensive understanding of the environment; to make the control system more resistant to interference, adaptable to uncertainty and external disturbances, and to provide safety for robotic arm control. Attached Figure Description

[0041] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the implementation of the invention and, together with the description, serve to explain the principles of the embodiments of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0042] Figure 1 This is a flowchart illustrating the nonlinear friction identification and compensation method for a robotic arm based on sliding mode control, as shown in the embodiments of this application.

[0043] Figure 2 This is a schematic diagram illustrating the friction identification and compensation of a rigid robotic arm joint with two degrees of freedom, as shown in an embodiment of this application.

[0044] Figure 3 This is a node friction diagram of a rigid robotic arm system with two degrees of freedom, as shown in an embodiment of this application.

[0045] Figure 4 This is a diagram showing the angular position of each link of a rigid robotic arm with two degrees of freedom, as illustrated in an embodiment of this application.

[0046] Figure 5 This is a diagram showing the angular velocity of each link of a rigid robotic arm with two degrees of freedom, as illustrated in an embodiment of this application.

[0047] Figure 6 This is a diagram showing the end effector position error of a rigid robotic arm with two degrees of freedom, as illustrated in an embodiment of this application. Detailed Implementation

[0048] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that the embodiments of the invention will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a full understanding of how embodiments of the invention are carried out.

[0049] Robotic arm control has broad significance, playing a crucial role in various fields and applications. Robotic arms are widely used in automated production lines in industrial manufacturing, improving production efficiency and quality. However, current robotic arm control algorithms still have some shortcomings. Some algorithms require substantial computational resources, making them unsuitable for embedded systems or resource-constrained environments. This increases system cost and complexity. Furthermore, robotic arm control algorithms may not be robust enough to external disturbances, sensor errors, or changes in the mechanical system, potentially leading to instability and degraded control performance. Some algorithms rely on predefined models and parameters, exhibiting poor adaptability to changing and unknown environments. A lack of sensing capabilities and real-time adjustment may limit the application of robotic arms in complex environments.

[0050] To address the aforementioned problems, this application provides a method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control, referring to... Figure 1 As shown, the method includes:

[0051] S100: Modeling yields the dynamic equations of an n-degree-of-freedom manipulator, which include the control input vector and friction compensation terms.

[0052]

[0053] In the formula, q(t) represents the position of the robotic arm, q¨(t) represents the velocity of the robotic arm, q˙(t) represents the acceleration of the robotic arm, λ represents the design constant of the robotic arm, S represents the state variable of the robotic arm, V represents the linear velocity of the robotic arm, and ω represents the angular velocity of the robotic arm.

[0054] in, The mass matrix represents the mass matrix, which is obtained based on the position of the robotic arm joints. G(q(t)) represents the Coriolis force matrix, which is obtained based on the velocity of the robotic arm joint; G(q(t)) represents the gravity vector, which is obtained based on the position of the robotic arm joint; u(t) represents the control input vector, which is the torque of the robotic arm joint motor.

[0055] The friction compensation term is represented as:

[0056]

[0057] In the formula, The friction torque loss parameter is... The difference between the actual measured friction torque values.

[0058] The control input vector is obtained through a nonlinear sliding mode control method, which can handle the nonlinear characteristics of the system. The friction compensation term is the difference between the friction torque loss parameter and the actual measured friction torque value. The friction compensation term is used to compensate for the losses caused by friction during the movement of the robotic arm.

[0059] In some embodiments of this application, the step of obtaining the control input vector through a nonlinear sliding mode control method further includes:

[0060] The control input vector, consisting of two equivalent symbols and one correction symbol, is obtained through a nonlinear sliding mode control method, and is defined as follows:

[0061]

[0062] In the formula, u(t) is the control input vector. For control gain or control matrix, For the system gain or parameter matrix, For the estimated friction compensation term, The product of the estimated mass matrix, the expected acceleration, and the measurement error. It is a nonlinear function term;

[0063] For the linear term of velocity V, N is the linear term for angular velocity ω, K is the matrix of gain coefficients, λ is the design constant, and N,K I ,R,K b This is the motor coefficient.

[0064] S200: The friction of the system is modeled using the Lugre friction model, resulting in the friction model equation. The friction model equation is simplified, and the nonlinear least squares method is used to estimate the model parameters.

[0065] The Lugre friction model is a mathematical model describing frictional forces, suitable for describing frictional forces with nonlinear characteristics. This model divides frictional force into static and dynamic frictional forces, and considers the influence of velocity and direction on friction. In the Lugre friction model, static and dynamic frictional forces are calculated using different formulas. Static friction is a nonlinear function that increases with relative velocity, reaching a saturation value when the velocity reaches a certain threshold. Dynamic friction, on the other hand, is a velocity-dependent function that decreases with increasing velocity. The advantage of the Lugre friction model lies in its ability to accurately describe the nonlinear characteristics of frictional forces and its capacity to handle frictional forces under different directions and velocities.

[0066] In some embodiments of this application, the friction of the system is modeled using the Lugre friction model, and the friction model equations are obtained by the following steps:

[0067] The friction model equation is obtained from the following formula:

[0068]

[0069] In the formula, F represents frictional force, and σ0z represents static frictional force. Represents kinetic friction. Represents damping force or resistance. Represents acceleration. Represents speed, Represents nonlinear terms;

[0070] in, for:

[0071]

[0072] In the formula, F c F represents the central force. S Represents saturation force, v s Represents linear velocity, σ v Represents angular velocity.

[0073] Furthermore, nonlinear least squares is a mathematical optimization technique used to solve the fitting problem of a set of nonlinear equations. This method fits experimental data to an empirical formula or theoretical function to minimize the sum of squared errors between predicted and actual values. The goal of nonlinear least squares is to adjust the model parameters to minimize the total residual. For a nonlinear model y = f(x, θ), where y is the system output, x is the input, and θ are the parameters (output, input, and parameters can be vectors), nonlinear least squares seeks the parameter estimates that minimize the sum of squared errors.

[0074] The simplified friction model equations in this application, and the estimation of the friction model equation parameters using the nonlinear least squares method, include the following steps:

[0075] By eliminating the internal acceleration variable in the friction model equation The friction model equations are simplified;

[0076] Assuming Estimating the parameters in the friction model equations as time-varying yields the following equation:

[0077]

[0078] The parameters are estimated using the nonlinear least squares method to obtain the final equation of the friction model. The parameters estimated by the nonlinear least squares method are expressed by the following formula:

[0079] (J T J+λI)Δθ=J T Δy;

[0080] In the formula, J represents a matrix, J T J represents the transpose matrix. T J represents the product of the transpose of J and J, called the covariance matrix of the error; λI represents the regularization term, where I is the identity matrix; Δθ represents the parameter vector to be estimated; Δy represents the measurement error.

[0081] The Lugre friction model can describe the nonlinear characteristics of friction. Simplifying the friction model equations makes them easier to handle. Furthermore, using the nonlinear least squares method to estimate the parameters of the friction model equations can find parameter values ​​that minimize the sum of squared errors between the data and the model predictions.

[0082] S300: Measure the output variables of the friction model equation, calculate the resulting torque based on the output variables, compare the resulting torque with the ideal input torque, and obtain the system friction torque loss parameter based on the comparison result; where the system friction torque loss parameter is a measure of the influence of system friction on the motion performance of the robotic arm.

[0083] Specifically, this application uses the torque difference method to calculate the system friction torque loss parameters. The torque difference method is a method for measuring the torque of a mechanical system. This method is based on the principle of a torque sensor, calculating the system torque by measuring the torque difference between two adjacent bearings. In this application, the system friction torque loss parameters are calculated using the torque difference method according to the following formula:

[0084]

[0085] In the formula, The system friction torque loss parameter, The input torque, u L (t) represents the load torque, which is the torque generated by other resistances or external forces in the system.

[0086] S400: By substituting the system friction torque loss parameters, the actual measured friction torque value, and the control input vector into the dynamic equations of the n-DOF manipulator, the dynamic equations of the manipulator after nonlinear friction identification and compensation are obtained. The equations consider the influence of friction and perform corresponding compensation, thus more accurately describing the motion behavior of the manipulator.

[0087] The method described in this application aims to accurately identify and compensate for nonlinear friction in a robotic arm, thereby improving its motion accuracy and stability. By combining nonlinear sliding mode control with the Lugre friction model, the influence of nonlinear frictional torque on the robotic arm during motion can be effectively addressed. Simultaneously, by measuring and comparing the system's frictional torque loss parameters, the impact of friction on the robotic arm's motion performance can be estimated more accurately, and compensation can be made accordingly. Finally, by incorporating the system's frictional torque loss parameters, the actual measured frictional torque value, and the control input vector into the dynamic equations, accurate identification and compensation for nonlinear friction in the robotic arm can be achieved, improving its motion performance and accuracy.

[0088] This application takes the joint friction identification and compensation of a two-degree-of-freedom robotic arm as an example, and verifies the technical effect of the method through simulation experiments. (Refer to...) Figure 2 As shown, according to the Lagrange equations, the general form of the robot's dynamics equations is as follows:

[0089]

[0090] The following simulation is performed using a two-degree-of-freedom planar robotic arm as an example, based on the schematic diagram and the dynamic parameters shown in the table below.

[0091] Table 1 Dynamic characteristics of the two robotic arms

[0092] parameter symbol numerical values Connecting rod mass (kg) <![CDATA[m1,m2]]> 0.5,0.5 <![CDATA[First link moment of inertia (kg·m 2 )]]> <![CDATA[I1,I2]]> 0.1,0.1 Linkage length (m) <![CDATA[a1,a2]]> 0.5,0.5

[0093] This application provides the friction model identification results at the joints of a robotic arm using two modes: a fixed-parameter friction model (CPFM) and a variable-parameter friction model (VPFM). (Refer to...) Figure 3 As shown, the friction values ​​for each node under the two modes and the friction maps obtained from the friction model were plotted and compared. Figure 3 As can be seen, the recognition error is large when the parameters remain unchanged, while the recognition error is small when the parameters remain unchanged. This clearly shows the impact of changes in friction parameters on recognition.

[0094] Reference Figure 4 and Figure 5 As shown, Figure 4 and Figure 5 The figures show the angular positions and angular velocities of each link under various friction compensation states. The comparison in the figures shows that the figure obtained using the time-varying parameter Lugre model for friction compensation is closer to the ideal system, and the system position error is smaller.

[0095] To prove the previous Figures 3-5 To verify the accuracy of the results, this application's embodiments plotted position error diagrams and made comparisons, referring to... Figure 6 As shown, in Figure 6 As can be seen from the graph, the graph with the variable parameter friction model is closer to the ideal system, and the system position error is also smaller.

[0096] The motion paths of the end effector under various states were plotted to better compare the performance of the proposed models. It can be seen that the path of the second type of model is closer to the ideal path of the robot, indicating that the final modified friction model has higher accuracy.

[0097] As described in the above embodiments, this application provides a method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control. The method includes: modeling an n-degree-of-freedom robotic arm's dynamic equations, which include a control input vector and friction compensation terms; modeling the system friction using the Lugre friction model to obtain friction model equations, simplifying the friction model equations, and estimating the model parameters using nonlinear least squares; measuring the output variables of the friction model equations and calculating the resulting torque based on the output variables, comparing the resulting torque with the ideal input torque, and obtaining system friction torque loss parameters based on the comparison results; and substituting the system friction torque loss parameters, the actual measured friction torque value, and the control input vector into the dynamic equations of the n-degree-of-freedom robotic arm to obtain the dynamic equations after nonlinear friction identification and compensation. This application, by implementing friction identification and compensation for the robotic arm, enhances the robotic arm's perception capabilities and further improves the accuracy of environmental perception and object recognition. Furthermore, it is necessary to improve perception fusion technology to integrate information from different sensors to provide a more comprehensive environmental understanding; make the control system more resistant to interference, adaptable to uncertainties and external disturbances, and provide safety for robotic arm control.

[0098] It should be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a structure, article, or apparatus that includes a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a structure, article, or apparatus. Without further limitations, an element defined by the phrase "comprising..." does not exclude the presence of other identical elements in the structure, article, or apparatus that includes the element.

[0099] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the disclosure herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the following claims.

Claims

1. A method for identifying and compensating nonlinear friction in a robotic arm based on sliding mode control, characterized in that, The methods include: The dynamic equations of the n-degree-of-freedom manipulator are obtained by modeling, and the dynamic equations of the n-degree-of-freedom manipulator include control input vectors and friction compensation terms; The control input vector is obtained through a nonlinear sliding mode control method, and the friction compensation term is the difference between the friction torque loss parameter and the actual measured friction torque value. The friction of the system is modeled using the Lugre friction model to obtain the friction model equation. The friction model equation is simplified, and the parameters of the friction model equation are estimated using the nonlinear least squares method. The output variables of the friction model equation are measured, and the resulting torque is calculated based on the output variables. The resulting torque is compared with the ideal input torque, and the system friction torque loss parameters are obtained based on the comparison results. Substituting the system friction torque loss parameters, the actual measured friction torque value, and the control input vector into the dynamic equation of the n-degree-of-freedom manipulator, the dynamic equation of the manipulator after nonlinear friction identification and compensation is obtained.

2. The method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control according to claim 1, characterized in that, The general form of the dynamic equations of the n-degree-of-freedom robotic arm is: In the formula, q(t) represents the position of the robotic arm, q ·· (t) represents the speed of the robotic arm, q · (t) represents the acceleration of the robotic arm, λ represents the design constant of the robotic arm, S represents the state variable of the robotic arm, V represents the linear velocity of the robotic arm, and ω represents the angular velocity of the robotic arm. in, The mass matrix represents the mass matrix, which is obtained based on the position of the robotic arm joints. G(q(t)) represents the Coriolis force matrix, which is obtained based on the velocity of the robotic arm joint; G(q(t)) represents the gravity vector, which is obtained based on the position of the robotic arm joint; u(T) represents the control input vector, which is the torque of the robotic arm joint motor. The friction compensation term is represented as: In the formula, The friction torque loss parameter is... This is the difference between the actual measured friction torque values.

3. The method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control according to claim 1, characterized in that, The step of obtaining the control input vector through the nonlinear sliding mode control method further includes: The control input vector, consisting of two equivalent symbols and one correction symbol, is obtained through a nonlinear sliding mode control method, and is defined as follows: In the formula, u(t) is the control input vector. For control gain or control matrix, For the system gain or parameter matrix, For the estimated friction compensation term, The product of the estimated mass matrix, the expected acceleration, and the measurement error. It is a nonlinear function term; For the linear term of velocity V, Let N be the linear term for angular velocity ω, K be the matrix of gain coefficients, λ be the design constant, and N, K be the linear term for angular velocity ω. I R, K b This is the motor coefficient.

4. The method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control according to claim 1, characterized in that, The steps for modeling system friction using the Lugre friction model and obtaining the friction model equations include: The friction model equation is obtained from the following formula: In the formula, F represents frictional force, and σ0z represents static frictional force. Represents kinetic friction. Represents damping force or resistance. Represents acceleration. Represents speed, Represents nonlinear terms; in, for: In the formula, F c F represents the central force. S σ represents saturation force, vs represents linear velocity, and σ represents saturation force. v Represents angular velocity.

5. The method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control according to claim 4, characterized in that, The steps of simplifying the friction model equations and estimating the friction model equation parameters using the nonlinear least squares method include: By eliminating the internal acceleration variables in the friction model equations The friction model equations are simplified; Assuming Estimating the parameters in the friction model equations as time-varying yields the following equation: The parameters are estimated using the nonlinear least squares method to obtain the final equation of the friction model. The parameters estimated by the nonlinear least squares method are expressed by the following formula: (J T J+λI)Δθ=J T Δy; In the formula, J represents a matrix, J T J represents the transpose matrix. T J represents the product of the transpose of J and J, called the covariance matrix of the error; λI represents the regularization term, where I is the identity matrix; Δθ represents the parameter vector to be estimated; Δy represents the measurement error.

6. The method for nonlinear friction identification and compensation of a robotic arm based on sliding mode control according to claim 1, characterized in that, The resulting torque is compared with the ideal input torque, and the system friction torque loss parameter is calculated based on the comparison result using the following formula: In the formula, The system friction torque loss parameter, The input torque, u L (t) represents the load torque, which is the torque generated by other resistances or external forces in the system.

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