A friction parameter identification method for a simplified multi-joint manipulator based on the Stribeck model

Abstract

The present invention discloses a simplified multi-joint manipulator friction parameter identification method based on the Stribeck model. The method process is as follows: the multi-joint manipulator system is split into multiple single-joint subsystems; a friction force model of each subsystem is constructed; for each subsystem, the driver is used to drive the subsystem to sample its angle, angular velocity, and torque; the sampled angular velocity is filtered, and the angular acceleration is obtained using a Butterworth filter; the angle, angular velocity, torque, and angular acceleration are substituted into the model of the single-joint subsystem to calculate the friction torque, and the friction parameters are identified based on the least squares method; the obtained friction parameters are used as the friction model parameters of the multi-joint manipulator system. The simplified multi-joint manipulator friction parameter identification method based on the Stribeck model proposed in the present invention can reduce the computational complexity of the multi-joint manipulator system friction parameter identification and can provide accurate modeling of the manipulator friction.

Description

Technical Field

[0001] The present invention relates to the technical field of robot dynamics parameter identification, in particular to a simplified multi-joint robot arm friction parameter identification method based on a Stribeck model. Background Art

[0002] In the fields of mechanical engineering and robotics, understanding and accurately modeling the frictional behavior of robotic arms is crucial for control system performance. Friction in robotic arms is caused by the motion of joints and includes various types of friction, such as dry friction, viscous friction, and Stribeck friction. Friction significantly affects the accuracy and control effectiveness of robotic arm motion. Therefore, accurately identifying the friction parameters of robotic arms is crucial for optimizing control algorithms and improving system performance.

[0003] The Stribeck model is a commonly used model for describing friction phenomena. It accounts for the nonlinear variation of friction with velocity and external forces. Therefore, there is an urgent need to accurately identify the friction parameters of robotic arm joints by combining experimental data with mathematical models, thereby improving the motion control performance of the robotic arm. Summary of the Invention

[0004] The purpose of the present invention is to solve the above-mentioned defects in the prior art and provide a simplified multi-joint robotic arm friction parameter identification method based on the Stribeck model. The complex multi-joint serial robotic arm system is split into multiple single-joint subsystems. By sampling the required relevant data, the parameters of the Stribeck model are fitted using the least squares method.

[0005] The purpose of the present invention can be achieved by taking the following technical solutions:

[0006] A simplified multi-joint robotic arm friction parameter identification method based on the Stribeck model, the parameter identification method comprising the following steps:

[0007] S1. Establish a dynamic model of the multi-joint robotic arm system based on the dynamic characteristics of the multi-joint robotic arm;

[0008] S2. Split the multi-joint robotic arm system into multiple single-joint subsystems and establish a Stribeck friction model for each single-joint subsystem;

[0009] S3, using a brushless DC motor driver to drive each joint of the robotic arm and sampling the encoder to obtain the angle, angular velocity and torque of each joint;

[0010] S4, filtering the angular velocity using a first-order low-pass filter, and low-pass filtering the angular velocity using a Butterworth filter to obtain angular acceleration;

[0011] S5. Based on the angle and torque collected in step S3 and the angular velocity and angular acceleration obtained by filtering in step S4, substitute them into the Stribeck friction model of the single-joint subsystem to calculate the friction torque of each single-joint subsystem;

[0012] S6. Based on the friction torque of the single joint subsystem, the friction parameters are identified using the least square method using MATLAB tools to obtain the parameters of the Stribeck friction model;

[0013] S7. Substitute the Stribeck friction model parameters obtained in step S6 into the dynamic model of the multi-joint robotic arm system in step S1 to obtain an accurate model of the multi-joint robotic arm system.

[0014] Furthermore, through the analysis and measurement of the actual multi-joint robotic arm system, combined with the description of the dynamic characteristics of the multi-joint robotic arm and the influence of external forces, a Lagrangian dynamic model of the multi-joint robotic arm system is established. The Lagrangian dynamic model is as follows:

[0015]

[0016] For an n-joint robot arm, θ=[θ1 ... θ n ] T is the n×1 dimensional joint angle vector, and are n×1 dimensional joint angular velocity and angular acceleration vectors respectively, M(θ) is the n×n dimensional mass matrix, is an n×n dimensional matrix, which contains the Coriolis force and centripetal force when the multi-joint manipulator system moves. g(θ) is an n×1 dimensional gravity vector, and τ is an n×1 dimensional torque vector. is an n×1 dimensional friction vector.

[0017] Furthermore, to address the effects of friction on the robotic arm during motion and improve the accuracy and stability of motion control, a Stribeck friction model for a single-joint subsystem was established. The Stribeck friction model accurately describes the nonlinear relationship between system friction and angular velocity. Furthermore, the Stribeck friction model has fewer parameters and lower computational complexity, making it effectively applicable to friction modeling in actual robotic arm systems. The Stribeck friction model is as follows:

[0018]

[0019] Where, v s is the characteristic velocity of the Stribeck friction model, F v is the viscous friction coefficient, F c is the Coulomb friction coefficient, Fs is the static friction coefficient.

[0020] Furthermore, the process of step S3 is as follows:

[0021] First, a brushless DC motor driver is used to perform simple motion control on a single-joint robotic arm. To ensure the reliability of the collected data, the system must operate stably under controlled conditions. Subsequently, an encoder installed on the drive motor monitors and records the motion process in real time over a certain period of time, collecting joint angle, angular velocity, and torque data at m sampling points.

[0022] Furthermore, the process of step S4 is as follows:

[0023] In order to eliminate data fluctuations caused by measurement noise and sampling errors and obtain smoother and more reliable angular velocity data, the angular velocity of the m sampling points collected is filtered. The processing method is as follows:

[0024] V new =k*V raw +(1-k)*V old

[0025] V raw is the angular velocity of the m sampling points obtained, k is the filter rate, the selection range is 0-1, V old is the filtered angular velocity of the previous sampling point, V new is the filtered angular velocity of the sampling point;

[0026] To preserve the low-frequency components in the filtered angular velocity data while suppressing high-frequency noise, a Butterworth filter is used to low-pass filter the angular velocity data to obtain angular acceleration information. The Butterworth filter is a commonly used digital filter characterized by a smooth frequency response curve, which minimizes the original signal's characteristics while filtering out high-frequency noise. This process can better capture the acceleration changes of a single joint subsystem. The low-pass filter formula is as follows:

[0027] H(s)=1 / (1+(s / jw c ) 2N )

[0028] H(s) is the transfer function of the filter, s is a complex frequency domain variable, jw c is the cutoff frequency, which controls the frequency response of the filter, and N is the order of the filter.

[0029] Furthermore, the process of step S5 is as follows:

[0030] Substitute the angles and torques of the m sampling points sampled in step S3 and the angular velocity and angular acceleration in step S4 into the Stribeck friction model of the single joint subsystem, and use the measured data to calculate and verify the friction model. According to the kinematic model of the single joint subsystem, the friction torque of the single joint subsystem can be calculated. as follows:

[0031]

[0032] Where: m is the mass of the single-joint subsystem, g is the acceleration due to gravity, l is the length of the robotic arm, and r is the joint operating point, corresponding to the distance from the motor position to the center of mass.

[0033] Furthermore, the process of step S6 is as follows:

[0034] Friction parameter identification is performed based on the least squares method. The least squares method is a mathematical optimization method that finds the best function matching of the data by minimizing the sum of squares of the errors and fitting the relationship between the independent variable and the dependent variable. First, the parameter set to be identified is defined as the independent variable. The parameter set to be identified for each single joint subsystem is defined as [F c ,F s ,v s ,F v ],

[0035] For each sampling point i, the observed value of the friction torque is is the dependent variable. i Defined as the observed value of the friction torque and the predicted value y of the friction torque i The differences between:

[0036] Define the residual sum of squares:

[0037] The partial derivative of the residual sum of squares for each parameter is calculated and the partial derivative is set to zero to obtain a system of equations. Finally, the system of equations is solved to obtain the optimal estimate of each parameter and the goodness of fit index is calculated to evaluate the model fitting effect. Adjust the parameter [F c ,F s ,v s ,F v ] Minimize the residual sum of squares to find the best fitting model and solve the equation to get:

[0038]

[0039] Furthermore, the process of step S7 is as follows:

[0040] First, the identified friction parameters for each joint are substituted into the Stribeck friction model to accurately describe the friction model for a single joint. Subsequently, the single-joint friction model is integrated into a multi-joint robotic arm system, resulting in a multi-joint robotic arm system that incorporates the Stribeck friction model. This model accounts for the effects of the interactions between joints and the overall motion on friction. By integrating the Stribeck friction model, the nonlinear characteristics of the system's friction as it changes with speed are described, allowing the system to better cope with variations in friction characteristics at different speeds.

[0041] The present invention has the following advantages and effects compared to the prior art:

[0042] The present invention uses the Stribeck friction model and the least squares method for parameter identification, thereby improving the accuracy of friction modeling in the manipulator dynamics model. Traditional multi-joint manipulator dynamics modeling methods do not take into account the impact of actual friction on manipulator control. The introduction of the Stribeck friction model and parameter identification improve the accuracy and robustness of manipulator motion in practical applications. In addition, friction models often have the problem of high model complexity. The present invention uses a simplified parameter identification method to split the multi-joint manipulator system into multiple single-joint subsystems, reducing the complexity of model identification and calculation. At the same time, the present invention performs two filtering operations on the sampled angular velocity information to obtain filtered angular velocity and angular acceleration, respectively, improving the quality of the sampled data, making it clearer, more stable, and more reliable, and avoiding the impact of interference or other unnecessary components on modeling accuracy during actual operation. The Stribeck model established by the present invention ensures an accurate description of the actual multi-joint manipulator system, making the multi-joint manipulator system containing the Stribeck model more consistent with the actual system, and having more practical guiding significance for the analysis, operation, and control of the multi-joint manipulator system. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0044] Figure 1 1 is a schematic flow chart of a simplified multi-joint robotic arm friction parameter identification method based on the Stribeck model provided in an embodiment of the present invention;

[0045] Figure 2 is a schematic diagram of a three-joint robotic arm according to an embodiment of the present invention;

[0046] Figure 3 3 is a flow chart of a process for processing the angular velocities of the 300 sampling points collected in step S3 according to an embodiment of the present invention;

[0047] Figure 4 is a fitting diagram of the Stribeck friction model of the three-degree-of-freedom serial type manipulator joint 1 in an embodiment of the present invention;

[0048] Figure 5 is a fitting diagram of the Stribeck friction model of the three-degree-of-freedom serial type manipulator joint 2 in an embodiment of the present invention;

[0049] Figure 6 is a fitting diagram of the Stribeck friction model of the three-degree-of-freedom serial type manipulator joint 3 in an embodiment of the present invention;

[0050] Figure 7 This is a trajectory tracking effect diagram of a multi-joint robotic arm system without adding a friction model according to an embodiment of the present invention using the calculated torque method;

[0051] Figure 8 This is a diagram showing the position trajectory tracking effect of a multi-joint robotic arm system with a Stribeck model added in an embodiment of the present invention using a calculated torque method. DETAILED DESCRIPTION

[0052] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0053] Example 1

[0054] Figure 1 FIG. 1 is a flow chart of a simplified multi-joint robotic arm friction parameter identification method based on the Stribeck model provided in an embodiment of the present invention. The simplified multi-joint robotic arm friction parameter identification method based on the Stribeck model includes the following steps:

[0055] S1. Establish a dynamic model of a multi-joint serial robot arm;

[0056] Figure 2 The following is a schematic diagram of a three-joint robotic arm. l1, l2, and l3 are the lengths of joints 1, 2, and 3, respectively. θ1 is the angle of joint 1 relative to the horizontal axis, and θ2 and θ3 are the angles of joints 2 and 3 relative to the previous joint. The Lagrangian dynamic model of this three-joint robotic arm is established as:

[0057]

[0058] Where θ = [θ1 θ2 θ3]T is a 3×1 dimensional joint angle vector, and are 3×1 dimensional joint angular velocity and angular acceleration vectors respectively, M(θ) is a 3×3 dimensional mass matrix, is a 3×3 dimensional matrix containing the Coriolis force and centripetal force during the motion of the multi-joint robotic arm system, g(θ) is a 3×1 dimensional gravity vector, and τ is a 3×1 dimensional torque vector. is a 3×1 dimensional friction vector.

[0059] S2. Split the multi-joint robotic arm system into multiple single-joint subsystems and establish a Stribeck friction model for each single-joint subsystem. The model is as follows:

[0060]

[0061] Where, v s is the characteristic velocity of the Stribeck friction model, F v is the viscous friction coefficient, F c is the Coulomb friction coefficient, F s is the static friction coefficient.

[0062] S3. Use a brushless DC motor driver to drive each single joint subsystem to perform simple movements. The encoder installed on the drive motor samples the angle, angular velocity, and torque of 300 sampling points within 3 seconds.

[0063] S4, Figure 3 This is the process of processing the angular velocity of the 300 sampling points collected in step S3. The angular velocity of the 300 sampling points collected is filtered using a first-order low-pass filter as follows:

[0064] V new =k*V raw +(1-k)*V old

[0065] V raw is the angular velocity of the 300 sampling points obtained, k is the filter rate, and the selection range is 0-1. In this embodiment, k is selected as 0.5, V old is the filtered angular velocity of the previous sampling point, V new is the filtered angular velocity of the sampling point;

[0066] The obtained filtered angular velocity is low-pass filtered using a Butterworth filter to obtain the angular acceleration, thereby retaining the low-frequency component and suppressing the high-frequency noise. The low-pass filtering formula is as follows:

[0067] H(s)=1 / (1+(s / jw c ) 2N )

[0068] H(s) is the transfer function of the filter, s is a complex frequency domain variable, jw c is the cutoff frequency, which is used to control the frequency response of the filter, and N is the order of the filter. In this embodiment, the sampling frequency is selected as 100 Hz, the low-pass cutoff frequency is selected as 10 Hz, and the order is selected as 4.

[0069] S5. Based on the angles and torques of the 300 sampling points in step S3 and the angular velocity and angular acceleration obtained by filtering in step S4, substitute them into the Stribeck friction model of the single-joint subsystem to calculate the friction torque of each single-joint subsystem. The calculation formula is as follows:

[0070]

[0071] Where: m is the mass of the single-joint subsystem, g is the acceleration due to gravity, l is the length of the robotic arm, and r is the joint operation point, which corresponds to the distance from the motor position to the center of mass.

[0072] S6. Based on the Stribeck friction model of the single-joint subsystem, the parameter set that needs to be identified for each single-joint subsystem is defined as [F c ,F s ,v s ,F v ].

[0073] For each sampling point i, the residual e i Defined as the observed value of the friction torque and the predicted value y of the friction torque i The differences between:

[0074] Define the residual sum of squares:

[0075] Perform the least squares estimation by adjusting the parameters [F c ,F s ,v s ,F v ] Minimize the residual sum of squares to find the best fitting model, and obtain the partial derivatives of the parameters separately and solve the equation:

[0076]

[0077] The above steps are implemented using MATLAB to identify the parameters for each joint friction model. Table 1 shows the parameters of the three-joint manipulator system and the Stribeck friction model identification results.

[0078] Table 1. Parameters of the three-joint manipulator system and identification results of the Stribeck friction model

[0079] Joint 1 Joint 2 Joint 3 m(g) 149.3 148 21.8 l(cm) 12.4 12.4 10.5 r(cm) 5.5 5.5 4.5 <![CDATA[F c ]]> -0.023881 0.020134 -0.051240 <![CDATA[F s ]]> -0.183500 0.020134 -0.051240 <![CDATA[v s ]]> 0.535241 0.042119 0.036506 <![CDATA[F v ]]> -0.055272 -0.349890 -0.135087

[0080] Figure 4-6 The following are the fitting curves of the Stribeck friction model for joints 1, 2, and 3 using the least squares method. It can be seen that the fitting curves of the three joints successfully describe the relationship between the angle and the sampling torque under the Stribeck friction model.

[0081] S7. Integrate the friction model of the above single joint subsystem into the three-joint robotic arm system to obtain a three-joint robotic arm system including the Stribeck friction model. The dynamic model of the three-joint robotic arm is then:

[0082]

[0083] Example 2

[0084] The Stribeck friction modeling method is applied to an actual multi-joint robotic arm system. The trajectory tracking effect of the multi-joint robotic arm system with and without the Stribeck friction model is compared to verify the effectiveness of the simplified multi-joint robotic arm friction identification method designed based on the Stribeck model. The dynamic model of the robotic arm without the friction model is:

[0085]

[0086] The dynamic model of the multi-joint robotic arm based on the Stribeck model is:

[0087]

[0088] For the robotic arm system studied in this example, the computed torque method (CTM) was used for trajectory tracking. The CTM is a method used to analyze and design robotic arm systems, relying on system modeling to achieve control of the robotic arm system. Specifically, when using the CTM to control the robotic arm system, the more accurate the system's dynamic model, the better the trajectory tracking effect and the smaller the error during trajectory tracking control.

[0089] The controller output of the calculated torque method is expressed as

[0090] u=Bu'+η

[0091] Where u is the controller output, u' is the linear controller output, B is the gain associated with the system model, and η is the feedback compensation associated with the system model. Assuming the system model parameters are completely known, the dynamics of the multi-joint manipulator without the friction model is:

[0092] B=M(θ)

[0093]

[0094] For the dynamics of a multi-joint manipulator based on the Stribeck model:

[0095] B=M(θ)

[0096]

[0097] The controller is designed to

[0098]

[0099] Where θ d is the desired joint angle vector, is the desired joint angular velocity vector, is the desired joint angular acceleration vector, K p , K d is a constant gain diagonal matrix. Let e ​​= θ d -θ is the angle error, so the two-part design can be combined to obtain:

[0100] Controller for the dynamics of a multi-joint manipulator without adding a friction model:

[0101]

[0102] The controller of the multi-joint manipulator dynamics based on the Stribeck model is:

[0103]

[0104] The closed-loop error equation of the system is:

[0105]

[0106] The controller parameters can be selected by configuring the dynamic response process to a critical damping state using the pole configuration method.

[0107] For a three-joint robotic arm, the controller parameters are selected as Kp=diag(4,10,10), Kd=diag(10,16,16), and the joint path points are generated using S-curve trajectory planning. The trajectory tracking effects before and after adding the Stribeck model are shown as follows: Figure 7 、 Figure 8 As shown in the figure, before the Stribeck model was added, the tracking performance of the three joints was poor, with some steady-state errors, especially the first joint. However, after the Stribeck model was added, all three joints achieved error-free tracking. The mean squared error (MSE) is defined to evaluate the tracking performance:

[0108]

[0109] Among them, θ i is the actual angle of the robot during movement, is the desired angle for trajectory planning.

[0110] A smaller mean square error indicates better tracking performance. Table 2 compares the mean square error before and after adding the Stribeck model. As can be seen from Table 2, the addition of the Stribeck model significantly reduces the mean square error of all three joints of the three-joint manipulator, significantly improving tracking performance compared to before the friction model was added.

[0111] Table 2. Comparison of mean square error before and after adding the Stribeck model

[0112]

[0113] The above embodiments are merely preferred embodiments for the purpose of fully illustrating the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are within the scope of protection of the present invention. The scope of protection of the present invention shall be subject to the claims.

Claims

1. A simplified multi-joint manipulator friction parameter identification method based on the Stribeck model, characterized in that: The parameter identification method comprises the following steps: S1. Establish a dynamic model of the multi-joint robotic arm system based on the dynamic characteristics of the multi-joint robotic arm; S2. Split the multi-joint robotic arm system into multiple single-joint subsystems and establish a Stribeck friction model for each single-joint subsystem; S3, using a brushless DC motor driver to drive each joint of the robotic arm and sampling the encoder to obtain the angle, angular velocity and torque of each joint; S4, filtering the angular velocity using a first-order low-pass filter, and low-pass filtering the angular velocity using a Butterworth filter to obtain angular acceleration; S5. Based on the angle and torque collected in step S3 and the angular velocity and angular acceleration obtained by filtering in step S4, substitute them into the Stribeck friction model of the single-joint subsystem to calculate the friction torque of each single-joint subsystem; S6. Based on the friction torque of the single joint subsystem, the friction parameters are identified using the least square method using MATLAB tools to obtain the parameters of the Stribeck friction model; S7. Substitute the Stribeck friction model parameters obtained in step S6 into the dynamic model of the multi-joint robotic arm system in step S1 to obtain an accurate model of the multi-joint robotic arm system.

2. The method for identifying friction parameters of a simplified multi-joint manipulator based on the Stribeck model according to claim 1, characterized in that: In step S1, a Lagrangian dynamics model of the multi-joint robotic arm is established according to the dynamic characteristics of the multi-joint robotic arm. The Lagrangian dynamics model is as follows: For an n-joint robotic arm, θ=[θ1 ... θ n ] T is the n×1 dimensional joint angle vector, and are n×1 dimensional joint angular velocity and angular acceleration vectors respectively, M(θ) is the n×n dimensional mass matrix, is an n×n dimensional matrix, which contains the Coriolis force and centripetal force when the multi-joint manipulator system moves. g(θ) is an n×1 dimensional gravity vector, and τ is an n×1 dimensional torque vector. is an n×1 dimensional friction vector.

3. The friction parameter identification method of a simplified multi-joint manipulator based on the Stribeck model according to claim 2, characterized in that: In step S2, a Stribeck friction model of a single joint subsystem is established, and the model is as follows: Where, v s is the characteristic velocity of the Stribeck friction model, F v is the viscous friction coefficient, F c is the Coulomb friction coefficient, F s is the static friction coefficient.

4. The method for identifying friction parameters of a simplified multi-joint manipulator based on the Stribeck model according to claim 3, characterized in that: The process of step S3 is as follows: The single-joint subsystem is driven by a brushless DC motor driver to perform simple motion, and the angle, angular velocity and torque of m sampling points within a certain time range are sampled by an encoder installed on the drive motor.

5. The method for identifying friction parameters of a simplified multi-joint manipulator based on the Stribeck model according to claim 4, characterized in that: The process of step S4 is as follows: The angular velocity of the m sampling points collected is filtered as follows: V new =k*V raw +(1-k)*V old V raw is the angular velocity of the m sampling points obtained, k is the filter rate, the selection range is 0-1, V old is the filtered angular velocity of the previous sampling point, V new is the filtered angular velocity of the sampling point; The obtained filtered angular velocity is low-pass filtered using a Butterworth filter to obtain the angular acceleration, thereby retaining the low-frequency component and suppressing the high-frequency noise. The low-pass filtering formula is as follows: H(s)=1 / (1+(s / jw c ) 2N ) H(s) is the transfer function of the filter, s is a complex frequency domain variable, jw c is the cutoff frequency, which controls the frequency response of the filter, and N is the order of the filter.

6. The method for identifying friction parameters of a simplified multi-joint manipulator based on the Stribeck model according to claim 5, characterized in that: The process of step S5 is as follows: Substitute the angles and torques of the m sampling points sampled in step S3 and the angular velocity and angular acceleration in step S4 into the Stribeck friction model of the single-joint subsystem to calculate the friction torque of the single-joint subsystem as follows: Where: m is the mass of the single-joint subsystem, g is the acceleration due to gravity, l is the length of the robotic arm, and r is the joint operating point, corresponding to the distance from the motor position to the center of mass.

7. The method for identifying friction parameters of a simplified multi-joint manipulator based on the Stribeck model according to claim 6, characterized in that: The process of step S6 is as follows: The friction parameters are identified based on the least square method, and the parameter set that needs to be identified for each single joint subsystem is defined as [F c ,F s ,v s ,F v ], For each sampling point i, the residual e i Defined as the observed value of the friction torque and the predicted value y of the friction torque i The differences between: Define the residual sum of squares: Perform the least squares estimation by adjusting the parameters [F c ,F s ,v s ,F v ] Minimize the residual sum of squares to find the best fitting model, by taking the partial derivative of each parameter separately and solving the equation to get:

8. The method for identifying friction parameters of a simplified multi-joint manipulator based on the Stribeck model according to claim 7, characterized in that: The process of step S7 is as follows: Substitute the identified friction parameters of each joint into the Stribeck friction model to obtain the friction model of the single joint subsystem; The friction model of the above single-joint subsystem is integrated into the multi-joint robotic arm system to obtain a multi-joint robotic arm system including the Stribeck friction model.

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