A method for identifying and suppressing friction vibration interference of a mechanical arm based on BLS-PSO

Abstract

The present application relates to a kind of BLS-PSO-based mechanical arm friction vibration interference identification and suppression method, it is applicable to the high-precision control of multi-degree-of-freedom flexible joint robot system.The method is first designed to meet the five-order Fourier excitation trajectory of zero boundary condition, to excite system dynamic response;Subsequently, based on Newton-Euler method, structural dynamics regression model is constructed, the minimum parameter set is extracted, and the output torque of structure item is obtained;For structural modeling residual, introduce breadth learning system (BLS) to nonlinear identification for non-structural disturbance, and adopt particle swarm optimization algorithm (PSO) to carry out global optimization to network structure parameter;Finally, structure item and disturbance item are fused, and total joint torque estimation is formed, for realizing feedforward compensation control.The present application has the advantages of high identification accuracy, strong real-time performance, high data utilization and excellent robustness, can effectively suppress the nonlinear friction torque disturbance generated by mechanical arm under vibration interference condition, improve the dynamic tracking performance and control stability of system, with good engineering adaptability and popularization value.

Description

Technical Field

[0001] This invention relates to the field of robot dynamics parameter identification and nonlinear disturbance compensation technology, specifically to a method for identifying and suppressing friction disturbances in a robotic arm based on vibration signals, which falls under the category of dynamic modeling and control accuracy improvement technology in intelligent manufacturing, industrial robot control, and intelligent sensing systems. Background Technology

[0002] With the rapid development of intelligent manufacturing and precision operation technologies, flexible joint robotic arms, with their lightweight structure and high-precision control capabilities, are widely used in complex work scenarios such as high-end assembly, micro-operation, and medical assistance. However, due to their high structural flexibility, significant joint friction, and tight drive-load coupling, robotic arm systems are highly susceptible to non-structural factors such as vibration disturbances, nonlinear friction, and load changes when performing rapid movements or frequently starting and stopping tasks. This leads to problems such as micro-vibration, large tracking errors, and response hysteresis, severely restricting control accuracy and dynamic stability.

[0003] Currently, mainstream dynamic identification and control methods mainly include least squares identification, recursive least squares, sliding mode control, feedforward compensation control, model predictive control (MPC), and nonlinear modeling methods based on deep neural networks. These methods can achieve basic system identification and compensation effects under certain conditions, but they often face the following challenges in practical applications: First, the models lack the ability to describe high-frequency disturbances and frictional hysteresis, making it difficult to capture subtle but crucial nonlinear dynamics; second, highly complex network models (such as BP and MLP) have long training times, are sensitive to initial conditions, and lack good online adaptive capabilities; third, some control methods do not fully utilize system excitation information, resulting in identification accuracy depending on a large number of data samples and low data utilization efficiency.

[0004] Therefore, in order to meet the requirements of nonlinear friction identification and real-time compensation of flexible robotic arms under vibration interference, there is an urgent need for an intelligent identification method with high identification accuracy, strong robustness and adaptability, which can accurately extract unstructured disturbance information under limited excitation data conditions and provide high-quality torque prediction and feedforward compensation support for the control system, thereby improving the tracking performance and system stability of the robotic arm in complex dynamic environments. Summary of the Invention

[0005] This invention aims to address the problems of inaccurate identification of non-structural disturbances, drastic changes in friction force, and low control compensation accuracy in existing robotic arm systems under vibration interference. It proposes a method for identifying and suppressing friction disturbances in robotic arms based on vibration signals. By integrating structural dynamics modeling and non-structural disturbance learning, it achieves high-precision identification and real-time compensation of nonlinear disturbance torques of joints under finite excitation conditions, thereby improving the dynamic response performance, trajectory tracking accuracy, and vibration robustness of flexible joint robotic arms.

[0006] To achieve the above objectives, the technical solution provided by this invention is as follows:

[0007] Step 1: For the target robotic arm joint, construct a fifth-order Fourier excitation trajectory q with zero boundary conditions. d,i (t), where q d,i (t) consists of several fundamental frequencies ω k The sine and cosine terms are superimposed, and the specific form is as follows:

[0008]

[0009] Where q d,i (t) represents the excitation displacement trajectory of the i-th joint; ω k Represented as the k-th sine or cosine fundamental frequency; a i,k and b i,k , where are the Fourier coefficients of the joints. All joint trajectories of the zero-boundary Fourier series satisfy the following constraints:

[0010] q d,i (0)=q d,i (T)=0

[0011]

[0012] Where: q d,i (T) represents the desired position of the i-th joint. This is represented as the corresponding expected velocity trajectory. The expected acceleration trajectory at the corresponding moment;

[0013] This constraint can avoid non-physical vibration excitation caused by discontinuous excitation, and at the same time, it constrains the displacement, velocity and acceleration to be zero at the start and end time points of the excitation cycle, so as to ensure the smoothness of the excitation input and the full excitation of the system response.

[0014] Step 2: Establish the structural dynamics equations of the robotic arm based on the Newton-Euler method, extract the minimum parameter set, and express the system dynamics expression in a linear regression form, specifically as follows:

[0015] τ=Y(x)·P+ε

[0016] Where τ is the joint torque, Y(x) is the regression matrix constructed from the state joints, and ε represents the unstructured disturbance residuals. q is the joint angle; For joint angular velocity and Joint angular acceleration;

[0017] Step 3: Based on the collected joint state information of the robotic arm and the output results of the established structural terms, construct an input-output dataset to describe the characteristics of unstructured disturbances. By establishing a BLS disturbance identification model, modeling and dynamic estimation of unstructured disturbance components such as joint friction, flexible interference and external excitation of the robotic arm are realized.

[0018] Step 4: PSO is used to globally optimize the structural parameters and regularization coefficients of the BLS perturbation model. By tracking the historical optimal position of individual particles and updating the state and the global optimal position of the group, it is used to make high-precision predictions and perturbation compensations for the actual torque of the system, thereby realizing the identification and suppression of friction interference of the robotic arm under the influence of vibration signals.

[0019] Step 5: The structural term output is fused with the non-structural disturbance estimation results to construct the total joint torque prediction value. This prediction value is then introduced into the controller as a feedforward compensation signal to achieve real-time suppression of friction disturbances in the robotic arm and improve control accuracy.

[0020] Furthermore, the BLS described in step 3, the establishment of its structure includes the following steps:

[0021] Step 3.1: The input layer of the BLS receives the state variables of the robotic arm joints. As inputs, these include joint angle q and angular velocity. With angular acceleration The feature mapping layer processes the input variables through several sets of random weights. and bias Perform linear transformations and nonlinear activations to generate feature vectors:

[0022]

[0023] Each Z (i) Provides a set of high-dimensional feature responses, which represent the horizontal concatenation of multiple sub-feature matrices to form a complete feature matrix, as shown in the following form:

[0024] Z = [Z (1) Z (2) ,...,Z (m) ]

[0025] Step 3.2: The enhancement layer of BLS further refines the feature map output Z. (i) An extended transformation is performed, and an enhancement vector H is obtained using another set of random weights. (j) The specific form is as follows:

[0026]

[0027] Where ξ (j)As the nonlinear activation function of the enhancement layer, the above mapping and enhancement process is completed in a non-iterative form, which improves the computational efficiency and online update capability of the model;

[0028] Step 3.3: The output layer of the BLS will output H from the enhancement layer. (j) Least square regression is performed on the target friction torque. By solving for the output weights, the error between the estimated output torque and the actual disturbance is minimized. A regularization term is introduced to suppress overfitting. The objective function is as follows:

[0029]

[0030] Where: τ i For the true output torque of the i-th sample, The model predicts the output torque for the i-th sample, where λ is the regularization coefficient used to control model complexity. This is the weight vector.

[0031] Furthermore, the PSO mentioned in step 4 is used to optimize the network structure parameters in BLS. These parameters include the number of feature mapping nodes, the number of augmentation nodes, the regularization coefficient, and the activation function type. The optimization process includes the following steps:

[0032] Step 4.1: In the initialization phase of BLS structure optimization, the particle swarm size N is set, and within the predefined range of each hyperparameter (such as the number of feature nodes, the number of enhancement nodes, and the regularization coefficient), the structure parameter vector and corresponding velocity vector of each particle are randomly generated to form the initial population. Subsequently, in each iteration, the particle dynamically adjusts its search path in the structure parameter space based on its historical optimal position (individual extreme value) and the global optimal position. The particle's position and velocity updates follow the following formula:

[0033]

[0034] in: p represents the BLS network structure parameters of the i-th particle after the k-th iteration. i,BLS Let g be the particle's current best historical position, and g be the historically best network structure parameters. The velocity vector for parameter updates, c1 and c2 are learning factors, w is a weight factor, and r1 and r2 are random terms;

[0035] Step 4.2: Further, a shrinkage factor χ is introduced to control the global convergence speed, and a dynamic inertial weighted annealing mechanism is adopted to improve the global search capability and local convergence stability; after each iteration, the fitness function is calculated based on the mean square error between the output torque of the BLS network and the actual frictional disturbance, and its objective function is:

[0036]

[0037] in: Set as the sum of individual cognitive factors and group social factors.

[0038] Step 4.3: To improve the boundary control capability of the particle swarm optimization process, initialize the particles:

[0039] x i (0)~U(x min ,x max ),v i (0)~U(v min ,v max )

[0040] Where: x i (0) represents the initial position vector of the i-th particle, v i (0) represents the initial velocity vector of the i-th particle, (x min ,x max ) represents the feasible region boundary of the particle position, and represents the upper and lower bounds of the robot arm's parameter space. min ,v max This limits the range of particle velocity.

[0041] The initial position and velocity of the particles are sampled uniformly within the upper and lower bounds to ensure global coverage of the search space and prevent particles from going out of bounds, which could lead to non-convergence. Position boundary conditions are set as follows:

[0042]

[0043] in: This indicates the position of particle i in the (t+1)th iteration. It also indicates that the updated position of the particle is less than the minimum value x. min Then the position stage will be x. min If the updated position of the particle is greater than the maximum value x max Cut it into x max Otherwise, retain the original updated value;

[0044] Step 4.4: Introduce a time-decay-based inertia factor annealing mechanism to achieve initial exploration and smooth convergence transition in the later stage. This is used to dynamically adjust the search weights of the particle PSO, improving the parameter optimization process and convergence efficiency. The specific expression is as follows:

[0045]

[0046] Where: ω(t) is the inertia weight for iteration number t, ω max The initial maximum inertia weight is used to control the breadth of the search in the early stages, ω minThe minimum inertia weight is used to control the accuracy in the later stages of convergence, T max This represents the maximum number of iterations.

[0047] Step 4.5: During the PSO optimization of BLS network parameters, a final convergence criterion is introduced. This criterion can prevent overfitting and improve training efficiency. Specifically, it takes the following form:

[0048] |J t -J t-1 |<ò

[0049] Among them: J t The objective function value represents the number of iterations, and ò is a preset convergence threshold. ω indicates that the fitness change is less than the threshold ò for two consecutive rounds, at which point the algorithm is considered to have reached a stable solution. The optimization process terminates, and the optimal structural parameters are output for BLS network construction, achieving optimal identification accuracy and robust compensation capability for friction disturbances under vibration signal conditions.

[0050] Furthermore, in step 5, the structural dynamics regression term is fused with the non-structural disturbance term output by BLS to construct the total torque output, thereby achieving high-precision estimation of the actual joint torque and effective compensation for vibration disturbances. This specifically includes the following steps:

[0051] Step 5.1: To enhance the model's fitting ability and control the accuracy of feedforward compensation, a non-structural perturbation term is introduced into the original structural model, resulting in the overall model expression as follows:

[0052] τ(t)=Y(x t )P+τ f (t)+e(t)

[0053] Where: τ(t) is the actual joint torque vector at the current moment, Y(x) t ) is the regression matrix, P is the structural dynamics parameter vector, and τ f e(t) represents the non-structural dynamic torque, and e(t) represents the measurement noise term.

[0054] Step 5.2: By combining the nonlinear modeling properties of BLS with the global optimization of PSO, the structural model terms are estimated using the least squares method, specifically in the following form:

[0055]

[0056] The specific expression for the residual is defined as follows:

[0057]

[0058] This expression defines the residual as the main source of non-structural disturbances. In order to prevent it from spreading in the control system and exciting structural vibrations, it needs to be modeled and suppressed.

[0059] Step 5.3: Construct the disturbance modeling function using BLS, specifically in the following form:

[0060]

[0061] in: F is the estimate of the unstructured disturbance at time t. BLS This represents the mapping function of BLS, where β is the connection weight of the BLS re-output layer;

[0062] Step 5.4: Combine the disturbance modeling function constructed by BLS with the residual definition to finally obtain the system identification model, which is expressed in the following form:

[0063]

[0064] in: This is the final estimated total exhalation torque. The torque is estimated based on the dynamic model of the robotic arm mechanism. This system identification model enables advance compensation for micro-vibration disturbances during the robotic arm motion control process, thereby improving the trajectory tracking accuracy, dynamic response consistency, and stability robustness of the overall system.

[0065] This invention proposes a method for identifying and suppressing frictional vibration interference in robotic arms based on BLS-PSO. Compared with existing technologies, its advantages include the following:

[0066] 1. High identification accuracy: Through the joint identification framework of structural dynamics and non-structural disturbance modeling, high-precision estimation of complex frictional disturbance torque can be achieved. In the simulation, the torque error can be controlled within 6×10-12 N·m.

[0067] 2. Strong real-time performance: The breadth learning system adopted has a shallow architecture and non-iterative training characteristics. Combined with the particle swarm optimization algorithm, it significantly improves the response speed and online adjustment capability of the recognition model.

[0068] 3. Strong robustness: The method can effectively cope with the influence of uncertain factors such as structural flexibility, parameter disturbance, and friction nonlinearity, and improve the motion stability of the robotic arm in a multi-disturbance environment;

[0069] 4. High data efficiency: Through efficient excitation trajectory design and low-sample learning mechanism, this method still has good convergence and prediction ability under limited training data conditions, which enhances its adaptability in practical industrial applications. Attached Figure Description

[0070] Figure 1 Flowchart of the method for identifying and suppressing frictional vibration interference of robotic arms based on BLS-PSO proposed in this invention;

[0071] Figure 2 The diagram shows the structure of a BLS-PSO-based method for identifying and suppressing frictional vibration interference in robotic arms.

[0072] Figure 3 Prediction of the frictional torque of joint 1 of the robot arm in Experiment Example 1;

[0073] Figure 4 Prediction of the frictional torque of joint 2 of the robot arm in Experiment Example 1;

[0074] Figure 5 Prediction of the frictional torque of joint 3 of the robot arm in Experiment Example 1;

[0075] Figure 6 Prediction of the frictional torque of joint 4 of the robot arm in Experiment Example 1;

[0076] Figure 7 Prediction of the frictional torque of joint 5 of the robot arm in Experiment Example 1;

[0077] Figure 8 Prediction of the frictional torque of joint 6 of the robot arm in Experiment Example 1;

[0078] Figure 9 The friction torque error of joint 1 of the robotic arm in Experiment Example 1;

[0079] Figure 10 The friction torque error of joint 2 of the robotic arm in Experiment Example 1;

[0080] Figure 11 The friction torque error of joint 3 of the robotic arm in Experiment Example 1;

[0081] Figure 12 The friction torque error of joint 4 of the robotic arm in Experiment Example 1;

[0082] Figure 13 The friction torque error of joint 5 of the robotic arm in Experiment Example 1;

[0083] Figure 14 The friction torque error of joint 6 of the robotic arm in Experiment Example 1;

[0084] Figure 15 The total torque output of joint 1 of the robotic arm in Experiment Example 1 is estimated;

[0085] Figure 16 The total torque output of joint 2 of the robotic arm in Experiment Example 1 is estimated;

[0086] Figure 17 The total torque output of joint 3 of the robotic arm in Experiment Example 1 is estimated;

[0087] Figure 18The total torque output of joint 4 of the robotic arm in Experiment Example 1 is estimated;

[0088] Figure 19 The total torque output of joint 5 of the robotic arm in Experiment Example 1 is estimated;

[0089] Figure 20 The total torque output of the robotic arm joint 6 in Experiment Example 1 is estimated;

[0090] Figure 21 This is a comparison of the three different methods in Experiment Example 1. Detailed Implementation

[0091] The present application will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present application, but do not limit the present application in any way. It should be noted that those skilled in the art can make several modifications and improvements without departing from the concept of the present application. These fall within the protection scope of the present application.

[0092] Example 1

[0093] This application discloses a method for identifying and suppressing frictional vibration interference in a robotic arm based on BLS-PSO, comprising the following steps:

[0094] Step 1: For the target robotic arm joint, construct a fifth-order Fourier excitation trajectory q with zero boundary conditions. d,i (t), where q d,i (t) consists of several fundamental frequencies ω k The sine and cosine terms are superimposed, and the specific form is as follows:

[0095]

[0096] Where q d,i (t) represents the excitation displacement trajectory of the i-th joint; ω k Represented as the k-th sine or cosine fundamental frequency; a i,k and b i,k , where are the Fourier coefficients of the joints. All joint trajectories of the zero-boundary Fourier series satisfy the following constraints:

[0097] q d,i (0)=q d,i (T)=0

[0098]

[0099] Where: q d,i (T) represents the desired position of the i-th joint. This is represented as the corresponding expected velocity trajectory. The expected acceleration trajectory at the corresponding moment;

[0100] This constraint can avoid non-physical vibration excitation caused by discontinuous excitation, and at the same time, it constrains the displacement, velocity and acceleration to be zero at the start and end time points of the excitation cycle, so as to ensure the smoothness of the excitation input and the full excitation of the system response.

[0101] Step 2: Establish the structural dynamics equations of the robotic arm based on the Newton-Euler method, extract the minimum parameter set, and express the system dynamics expression in a linear regression form, specifically as follows:

[0102] τ=Y(x)·P+ε

[0103] Where τ is the joint torque, Y(x) is the regression matrix constructed from the state joints, and ε represents the unstructured disturbance residuals. q is the joint angle; For joint angular velocity and Joint angular acceleration;

[0104] Step 3: Based on the collected joint state information of the robotic arm and the output results of the established structural terms, construct an input-output dataset to describe the characteristics of unstructured disturbances. By establishing a BLS disturbance identification model, modeling and dynamic estimation of unstructured disturbance components such as joint friction, flexible interference and external excitation of the robotic arm are realized.

[0105] Step 4: PSO is used to globally optimize the structural parameters and regularization coefficients of the BLS perturbation model. By tracking the historical optimal position of individual particles and updating the state and the global optimal position of the group, it is used to make high-precision predictions and perturbation compensations for the actual torque of the system, thereby realizing the identification and suppression of friction interference of the robotic arm under the influence of vibration signals.

[0106] Step 5: The structural term output is fused with the non-structural disturbance estimation results to construct the total joint torque prediction value. This prediction value is then introduced into the controller as a feedforward compensation signal to achieve real-time suppression of friction disturbances in the robotic arm and improve control accuracy.

[0107] Furthermore, the BLS described in step 3, the establishment of its structure includes the following steps:

[0108] Step 3.1: The input layer of the BLS receives the state variables of the robotic arm joints. As inputs, these include joint angle q and angular velocity. With angular acceleration The feature mapping layer processes the input variables through several sets of random weights. and bias Perform linear transformations and nonlinear activations to generate feature vectors:

[0109]

[0110] Each Z (i) Provides a set of high-dimensional feature responses, which represent the horizontal concatenation of multiple sub-feature matrices to form a complete feature matrix, as shown in the following form:

[0111] Z = [Z (1) Z (2) ,...,Z (m) ]

[0112] Step 3.2: The enhancement layer of BLS further refines the feature map output Z. (i) An extended transformation is performed, and an enhancement vector H is obtained using another set of random weights. (j) The specific form is as follows:

[0113]

[0114] Where ξ (j) As the nonlinear activation function of the enhancement layer, the above mapping and enhancement process is completed in a non-iterative form, which improves the computational efficiency and online update capability of the model;

[0115] Step 3.3: The output layer of the BLS will output H from the enhancement layer. (j) Least square regression is performed on the target friction torque. By solving for the output weights, the error between the estimated output torque and the actual disturbance is minimized. A regularization term is introduced to suppress overfitting. The objective function is as follows:

[0116]

[0117] Where: τ i For the true output torque of the i-th sample, The model predicts the output torque for the i-th sample, where λ is the regularization coefficient used to control model complexity. This is the weight vector.

[0118] Furthermore, the PSO mentioned in step 4 is used to optimize the network structure parameters in BLS. These parameters include the number of feature mapping nodes, the number of augmentation nodes, the regularization coefficient, and the activation function type. The optimization process includes the following steps:

[0119] Step 4.1: In the initialization phase of BLS structure optimization, the particle swarm size N is set, and within the predefined range of each hyperparameter (such as the number of feature nodes, the number of enhancement nodes, and the regularization coefficient), the structure parameter vector and corresponding velocity vector of each particle are randomly generated to form the initial population. Subsequently, in each iteration, the particle dynamically adjusts its search path in the structure parameter space based on its historical optimal position (individual extreme value) and the global optimal position. The particle's position and velocity updates follow the following formula:

[0120]

[0121] in: p represents the BLS network structure parameters of the i-th particle after the k-th iteration. i,BLS Let g be the particle's current best historical position, and g be the historically best network structure parameters. The velocity vector for parameter updates, c1 and c2 are learning factors, w is a weight factor, and r1 and r2 are random terms;

[0122] Step 4.2: Further, a shrinkage factor χ is introduced to control the global convergence speed, and a dynamic inertial weighted annealing mechanism is adopted to improve the global search capability and local convergence stability; after each iteration, the fitness function is calculated based on the mean square error between the output torque of the BLS network and the actual frictional disturbance, and its objective function is:

[0123]

[0124] in: Set as the sum of individual cognitive factors and group social factors.

[0125] Step 4.3: To improve the boundary control capability of the particle swarm optimization process, initialize the particles:

[0126] x i (0)~U(x min ,x max ),v i (0)~U(v min ,v max )

[0127] Where: x i (0) represents the initial position vector of the i-th particle, v i (0) represents the initial velocity vector of the i-th particle, (x min ,x max ) represents the feasible region boundary of the particle position, and represents the upper and lower bounds of the robot arm's parameter space. min ,v max This limits the range of particle velocity.

[0128] The initial position and velocity of the particles are sampled uniformly within the upper and lower bounds to ensure global coverage of the search space and prevent particles from going out of bounds, which could lead to non-convergence. Position boundary conditions are set as follows:

[0129]

[0130] in: This indicates the position of particle i in the (t+1)th iteration. It also indicates that the updated position of the particle is less than the minimum value x. min Then the position stage will be x. min If the updated position of the particle is greater than the maximum value x max Cut it into x max Otherwise, retain the original updated value;

[0131] Step 4.4: Introduce a time-decay-based inertia factor annealing mechanism to achieve initial exploration and smooth convergence transition in the later stage. This is used to dynamically adjust the search weights of the particle PSO, improving the parameter optimization process and convergence efficiency. The specific expression is as follows:

[0132]

[0133] Where: ω(t) is the inertia weight for iteration number t, ω max The initial maximum inertia weight is used to control the breadth of the search in the early stages, ω min The minimum inertia weight is used to control the accuracy in the later stages of convergence, T max This represents the maximum number of iterations.

[0134] Step 4.5: During the PSO optimization of BLS network parameters, a final convergence criterion is introduced. This criterion can prevent overfitting and improve training efficiency. Specifically, it takes the following form:

[0135] |J t -J t-1 |<ò

[0136] Among them: J t The objective function value represents the number of iterations, and ò is a preset convergence threshold. ω indicates that the fitness change is less than the threshold ò for two consecutive rounds, at which point the algorithm is considered to have reached a stable solution. The optimization process terminates, and the optimal structural parameters are output for BLS network construction, achieving optimal identification accuracy and robust compensation capability for friction disturbances under vibration signal conditions.

[0137] Furthermore, in step 5, the structural dynamics regression term is fused with the non-structural disturbance term output by BLS to construct the total torque output, thereby achieving high-precision estimation of the actual joint torque and effective compensation for vibration disturbances. This specifically includes the following steps:

[0138] Step 5.1: To enhance the model's fitting ability and control the accuracy of feedforward compensation, a non-structural perturbation term is introduced into the original structural model, resulting in the overall model expression as follows:

[0139] τ(t)=Y(x t )P+τ f (t)+e(t)

[0140] Where: τ(t) is the actual joint torque vector at the current moment, Y(x) t ) is the regression matrix, P is the structural dynamics parameter vector, and τ f e(t) represents the non-structural dynamic torque, and e(t) represents the measurement noise term.

[0141] Step 5.2: By combining the nonlinear modeling properties of BLS with the global optimization of PSO, the structural model terms are estimated using the least squares method, specifically in the following form:

[0142]

[0143] The specific expression for the residual is defined as follows:

[0144]

[0145] This expression defines the residual as the main source of non-structural disturbances. In order to prevent it from spreading in the control system and exciting structural vibrations, it needs to be modeled and suppressed.

[0146] Step 5.3: Construct the disturbance modeling function using BLS, specifically in the following form:

[0147]

[0148] in: F is the estimate of the unstructured disturbance at time t. BLS This represents the mapping function of BLS, where β is the connection weight of the BLS re-output layer.

[0149] Step 5.4: Combine the disturbance modeling function constructed by BLS with the residual definition to finally obtain the system identification model, which is expressed in the following form:

[0150]

[0151] in: This is the final estimated total exhalation torque. The torque is estimated based on the dynamic model of the robotic arm mechanism. This system identification model enables advance compensation for micro-vibration disturbances during the robotic arm motion control process, thereby improving the trajectory tracking accuracy, dynamic response consistency, and stability robustness of the overall system.

[0152] The following simulation demonstrates the effectiveness and feasibility of the control strategy based on BLS-PSO for identifying and suppressing frictional vibration disturbances in a robotic arm, as disclosed in this application. The simulation details and results are as follows:

[0153] The parameters for the simulation experiment are designed as follows:

[0154] System parameter settings: Sampling frequency: 1kHz, total simulation time: 20s, joint inertia: J = 0.03kg·m 2 The friction torque model considers Coulomb friction and viscous friction, where: Coulomb friction coefficient: τ c =0.5Nm, viscous friction coefficient b = 0.01Nm·s / rad, external disturbance: add a 5Hz sinusoidal disturbance d(t) = 0.1sin(2π·5t) to simulate periodic disturbance. BLS system parameter settings: number of feature mapping nodes: 40, number of enhancement nodes: 25, regularization coefficient λ: 0.001;

[0155] PSO parameter settings: number of particles: 30, maximum number of iterations: 100, inertia factor: ω decreases linearly from 0.9 to 0.4, learning factor: c1 = c2 = 2.0.

[0156] Simulation results are as follows Figures 3-21 As shown in the figure. For each joint of the six-DOF robotic arm, the actual frictional torque and the frictional disturbance output predicted by the BLS network are extracted and compared. This analysis aims to verify whether the BLS network can effectively filter out high-frequency noise and extract meaningful frictional torque under actual control conditions, while evaluating whether the accuracy of the frictional torque prediction meets the requirements of the control system.

[0157] like Figures 3-8 As shown, for each joint of a six-DOF robotic arm, the actual torque extracted and the predicted frictional disturbance output by the BLS network are compared and analyzed. This verifies whether the BLS network can filter out high-frequency noise and extract meaningful frictional torque under actual control, and also verifies whether the accuracy of the frictional force meets the control requirements.

[0158] Through observation Figure 3 Throughout the entire motion cycle, the friction torque curve output by BLS at the first joint can fit the ideal curve well in terms of trend and amplitude. Especially in the low-speed section (2s-4s) and near the speed reversal point, BLS shows a keen response to nonlinear disturbance dynamics. Particularly at the two speed reversal nodes at the 6th and 18th seconds, the friction curve shows abrupt changes, and the BLS system successfully fits it through residual learning.

[0159] Joint 2 Figure 4In the model, the BLS network output and the reference friction curve maintain a high degree of consistency in amplitude and phase. Especially between 5 and 10 seconds, when the system is in a stable motion range, the BLS prediction of frictional disturbances almost overlaps with the actual curve, demonstrating an effective fitting ability for velocity-related frictional components. At several high-frequency vibration disturbance nodes during the motion of the second joint (such as t=3s), the frictional torque exhibits typical nonlinear abrupt changes (such as stick-slip effect or Stribeck region disturbance). The model is still able to maintain a rapid response to these disturbance trends without significant delay or fuzzy fluctuations.

[0160] Joint 3 Figure 5 Compared to the previous two joints, the friction output is more complex, exhibiting strong asymmetry and velocity-dependent coupling characteristics. For example, during the commutation phase at t=6s, the friction torque waveform not only shows a significant abrupt change in slope but also exhibits a noticeable "peak-valley" effect. The friction model demonstrates strong dynamic tracking capabilities, successfully capturing the friction disturbance mode of this joint under low-speed, high-load conditions. Furthermore, the predicted curve and the reference curve are highly aligned at multiple "disturbance response inflection points."

[0161] Joint 4 Figure 6 It exhibits more typical low-speed, high-viscosity friction characteristics, with smaller fluctuations in frictional torque amplitude. However, it displays a nonlinear "hysteresis response" in the initial stage of startup and the region of slight speed variation. In particular, around t=3s and t=10.5s, the friction curve shows a smooth but nonlinearly accelerated change. The constructed friction model can still maintain the stability of tracking the trend of weak disturbances. Its output blue line does not show any offset or delay at the above key points, exhibiting continuous, smooth, and well-fitted characteristics.

[0162] Figure 7 For predicting the friction torque of the fifth joint, the system alternately performs fast and slow rhythm switching tasks at different frequency bands. The friction torque is affected by both speed and load changes, and the disturbance exhibits strong non-stationarity and mid-frequency dynamic offset characteristics. From Figure 7 The results show that the model curve is generally stable with minimal delay throughout the entire motion cycle. It exhibits stable tracking ability at velocity jump nodes (such as t=6s and t=12.2s) and can still maintain good fit in sections with large friction change slopes in multiple time periods (such as t=7.5-9.0s).

[0163] Joint 6 Figure 8The settings are biased towards the system edge channels, with the maximum torque only within ±0.04 N·m, but the perturbation frequency is relatively high, exhibiting a slight jitter and fluctuation pattern. Image results show that although the friction fluctuation amplitude is large, BLS can capture the main rhythm and peak trends in multiple high-frequency bands, especially at the start and end points of the cycle, without showing obvious drift, demonstrating good fitting results and verifying the stability of this modeling in small-signal modeling.

[0164] Figure 9 The torque prediction error for joint 1 is shown. The error remains within ±0.008 N·m and exhibits slight asymmetry. A peak deviation of +0.05 N·m occurs at the acceleration transition point (t≈2.1 s), while a negative peak of -0.06 N·m appears near the velocity reversal point (t≈8.7 s). During the steady motion phase (t=4–7 s), the error tends to stabilize. These results confirm the ability of the BLS model to maintain continuity and dynamic response at motion inflection points.

[0165] Figure 10 Error modes for joint 2 are shown, exhibiting periodic oscillations of approximately ±0.018 N·m at t≈3.5 seconds. The frequency corresponds to the second-order bending mode of the robotic arm, indicating the presence of flexible coupling disturbances. Nevertheless, the BLS network effectively constrains the residual vibrational energy, demonstrating its suitability for flexible joint modeling and dynamic decoupling.

[0166] Figure 11 The error characteristics of joint 3 are shown, with deviations consistently limited to ±0.015 N·m. Notable peaks occur at t≈2.1 s and 8.7 s, corresponding to nonlinear transient phenomena in tribodynamics (such as the Stribeck effect). The BLS framework accurately captures these nonlinear features while maintaining overall convergence, reflecting a strong sensitivity to hysteresis effects and transient dynamics.

[0167] Figure 12 The error at joint 4 was shown, exhibiting higher frequency fluctuations with a maximum amplitude of ±0.12 N·m. These oscillations may originate from the meshing dynamics of the harmonic reducer. Despite the challenging conditions, the regularization parameter (λ = 1 × 10⁻⁶) tuned by PSO was... -2 This ensures stable output and suppresses the amplification of low torque noise, highlighting the robustness of the model.

[0168] Figure 13 The prediction error for joint 5 is shown. The error signal is dense and irregular, with peak values ​​close to ±0.013 N·m, mainly concentrated in the frequency band related to mechanical meshing. By applying an extended BLS structure with 20 enhancement nodes, the network effectively attenuates transient disturbances, demonstrating its effectiveness in handling non-stationary structural noise.

[0169] Figure 14 The error characteristics of joint 6 are shown, displaying a composite signal consisting of white noise and low-amplitude periodic components. The error stabilizes within ±0.015 N·m without significant drift. These characteristics demonstrate the strong filtering capability of the BLS model under complex perturbation conditions such as commutation noise and joint flexibility.

[0170] Figure 15-20 The diagram shows a comparison between the ideal total torque curve (blue line) and the total torque output predicted by BLS (red line) across all six joints.

[0171] Joint 1 as Figure 15 As shown, the two curves exhibit a high degree of overall alignment, almost completely overlapping within the main motion region (e.g., from t = 5 s to t = 10 s), indicating that the overall model effectively captures the system's inertial and frictional characteristics. Slight deviations occur during the start-stop phases (approximately t = 0–2 s and t = 9–10 s), which can be attributed to the sudden coupling effect between the system's inertia and friction—a behavior that falls within the expected range of the standard identification model.

[0172] Figure 16 The comparison curves for joint 2 show good overall performance. In particular, during the steady-state motion phase between 7 and 9 seconds, the predicted torque is very close to the ideal torque, demonstrating strong predictive capability. Near the two significant speed reversal points (approximately at t = 4 seconds and t = 9 seconds), the predicted torque exhibits a slight hysteresis, with a maximum error of approximately 0.04 N·m. This is due to the coupling between dynamic inertia changes and the unsteady characteristics of friction. This performance is consistent with and within a reasonable range of the inertial delay effect typically observed in nonlinear dynamic system modeling.

[0173] exist Figure 17 In the comparison of the total torque at joint 3, more complex nonlinear characteristics are observed. The predicted curve (blue line) generally follows the trend of the ideal curve (red line), but significant fluctuations appear in the predicted torque during several high-frequency motion phases (e.g., t = 7–10 seconds). These fluctuations are attributed to frequent rate reversals and load changes in joint 3, reflecting its complex coupled perturbation dynamics. Nevertheless, the overall prediction error remains within ±0.045 N·m, highlighting the model's sensitivity to unstructured perturbations.

[0174] exist Figure 18 In the test, the torque prediction for joint 4 was exceptionally stable. Due to the low intensity of the nonlinear disturbance and the more consistent friction-inertia relationship of this joint, the two curves exhibited extremely high alignment. The overall error remained within ±0.02 N·m, with almost no significant fluctuations.

[0175] Figure 19 The results show that the total torque comparison at joint 5 exhibits significant high-frequency disturbance characteristics. The predicted curve generally follows the trend of the ideal curve for most of the motion cycle, especially performing exceptionally well during the high-speed transition phase (e.g., around t=6s and t=9s). However, a slight deviation of approximately 0.03 N·m is observed at specific local points (e.g., t=3s).

[0176] for Figure 20 As shown at node 6, the total torque prediction exhibits frequent disturbances and irregular fluctuations. Although the prediction curve shows obvious random behavior, it closely matches the overall trend of the ideal total torque. The error distribution is uniform, with the maximum deviation remaining below 0.02 N·m, indicating that the system has a strong ability to track random disturbances.

[0177] To compare and evaluate the torque prediction performance of different modeling methods, Figure 21 The image shows comparisons between the predictions and actual values ​​of three representative methods: RLS, MLP, and the proposed BLS-PSO. From top to bottom, each subplot corresponds to a comparison between the predicted and actual values ​​of RLS, MLP, and BLS-PSO, respectively.

[0178] RLS (top): Although the Recursive Least Squares (RLS) method can capture the overall trend of the torque curve, it exhibits significant deviations in regions with sharp fluctuations. This indicates its limited adaptability to the nonlinear and non-smooth characteristics of frictional torque;

[0179] MLP (Intermediate Layer): Multilayer perceptrons (MLPs) outperform RLS in capturing high-frequency variations. However, in certain specific regions, especially when the signal is noisy or dynamically changing, it still suffers from overshoot and irregular fitting issues.

[0180] BLS-PSO (bottom): In comparison, the proposed BLS-PSO method demonstrates significantly improved prediction accuracy, with its output curve closely following the real torque across the entire sample exponential range. The alignment between the green predicted line and the black true curve is noticeably tighter, validating the effectiveness of combining BLS with PSO-based adaptive optimization in handling nonlinearity and model uncertainty. This comparative result highlights the superior generalization and noise robustness of the proposed method, making it a more suitable choice for real-time robot torque modeling applications.

[0181] The technical means disclosed in this invention are not limited to those disclosed in the above embodiments, but also include technical solutions that are any combination of the above technical features. It should be noted that for those skilled in the art, various improvements and modifications can be made without departing from the principle of this invention, and these modifications are also considered within the scope of protection of this invention.

Claims

1. A method for identifying and suppressing frictional vibration interference in a robotic arm based on BLS-PSO, characterized in that, include: Step 1: For the target robotic arm joint, construct a fifth-order Fourier excitation trajectory with zero boundary conditions; Step 2: Establish the structural dynamics equations of the robotic arm based on the Newton-Euler method, extract the minimum parameter set, and express the system dynamics expression in a linear regression form; Step 3: Based on the collected joint state information of the robotic arm and the output results of the established structural terms, construct an input-output dataset to describe the characteristics of unstructured disturbances. By establishing a Breadth-Layered Learning System (BLS) disturbance identification model, modeling and dynamic estimation of the unstructured disturbance components of the robotic arm joint friction, flexible disturbances, and external excitations are realized. This includes: introducing the BLS system to perform nonlinear identification of unstructured disturbances in the mechanical structure regression residuals; the BLS network takes the joint state variables of the robotic arm as input, including position, velocity, and acceleration, and the friction torque as output, to construct the mapping relationship between joint state and nonlinear disturbances. Step 4: The Particle Swarm Optimization (PSO) algorithm is used to globally optimize the structural parameters of the BLS perturbation identification model. By tracking the historical optimal position of individual particles and updating the state and the global optimal position of the group, the frictional interference of the robotic arm under the influence of vibration signals is identified and suppressed. The structural parameters include the number of feature mapping nodes, the number of enhancement nodes, the regularization coefficient, and the activation function type. The optimization process includes: Step 4.1: Initialization phase. Set the particle swarm size N, and randomly generate the structure parameter vector and velocity vector for each particle within the predefined range of each hyperparameter to form the initial population. In each iteration, the particle dynamically adjusts its search path in the structure parameter space based on its historical best position and global best position. The particle's position and velocity are updated as follows: ； ； in: p represents the structure parameters of the i-th particle after the k-th iteration. i,BLS denoted as the current best historical position of the particle, g is the globally best structure parameter in history, c1 and c2 are learning factors, w is the weight factor, and r1 and r2 are random terms; Step 4.2: Introduce a shrinkage factor to control the global convergence speed and adopt a dynamic inertial weighted annealing mechanism to improve the global search capability and local convergence stability; after each iteration, calculate the fitness function based on the mean square error between the output torque of the BLS network and the actual frictional disturbance. Step 4.3: To improve the boundary control capability of the particle swarm optimization process, initialize the particles; Step 4.4: Introduce a time-decay-based inertia factor annealing mechanism to achieve initial exploration and smooth convergence transition in the later stage. This is used to dynamically adjust the search weight of PSO and improve the parameter optimization process and convergence efficiency. Step 4.5: During the optimization process, introduce the final convergence criterion; Step 5: The structural term output is fused with the non-structural disturbance estimation results to construct the total joint torque prediction value. The prediction value is then introduced into the controller as a feedforward compensation signal to achieve real-time suppression of friction disturbances in the robotic arm and improve control accuracy.

2. The method for identifying and suppressing frictional vibration interference in a robotic arm based on BLS-PSO according to claim 1, wherein: Step 1: For the target robotic arm joint, construct a fifth-order Fourier excitation trajectory with zero boundary conditions, consisting of the superposition of several fundamental frequency sine and cosine terms, specifically in the form of: ； in Representing the The excitation displacement trajectory of each joint; Represented as the first A sine or cosine fundamental frequency; and Let be the Fourier coefficients of the joints, and be the zero-boundary Fourier series. All joint trajectories satisfy the following constraints: ； ； ； The constraints are used to avoid non-physical vibration excitation caused by discontinuous excitation, and at the same time, the displacement, velocity and acceleration at the start and end time points of the excitation cycle are constrained to be zero, so as to ensure the smoothness of the excitation input and the full excitation of the system response. Step 2: Establish the structural dynamics equations of the robotic arm based on the Newton-Euler method, extract the minimum parameter set, and express the system dynamics expression in a linear regression form, specifically as follows: τ=Y(x)·P+ε; Where τ is the joint torque, Y(x) is the regression matrix constructed from the joint state, P is the structural dynamics parameter vector, and ε represents the non-structural disturbance residual. q is the joint angle; For joint angular velocity and This is the joint angular acceleration.

3. The method for identifying and suppressing frictional vibration interference in a robotic arm based on BLS-PSO as described in claim 2, characterized in that, The structure establishment of the BLS described in step 3 includes the following steps: Step 3.1: The input layer of the BLS receives the state variables of the robotic arm joints. As inputs, these include joint angle q and joint angular velocity. With joint angular acceleration The feature mapping layer processes the input variables through several sets of random weights. and bias Perform linear transformations and nonlinear activations to generate feature vectors: ； Each Z (i) Provide a set of high-dimensional features; A complete feature matrix is ​​formed, in the following form: Z=[Z (1) ,WITH (2) ,...,WITH (m) ]; Step 3.2: The enhancement layer of the BLS outputs Z on the feature map. (i) An extended transformation is performed, and an enhancement vector H is obtained using another set of random weights. (j) The specific form is as follows: ； Where ξ (j) The nonlinear activation function for the enhancement layer is used, and the mapping and enhancement process is completed in a non-iterative manner, which improves the computational efficiency and online update capability of the model. Step 3.3: The output layer of the BLS will output H from the enhancement layer. (j) Least square regression is performed on the target friction torque. By solving for the output weights, the error between the estimated output torque and the actual disturbance is minimized. A regularization term is introduced to suppress overfitting. The objective function is as follows: ； in: For the first The true output torque of each sample For the first The model predicts the output torque for each sample, where λ is the regularization coefficient used to control model complexity. This is the weight vector.

4. The method for identifying and suppressing frictional vibration interference in a robotic arm based on BLS-PSO as described in claim 2, characterized in that, Step 4 includes: Step 4.2: Introduce a shrinkage factor to control the global convergence speed and employ a dynamic inertial weighted annealing mechanism to improve global search capability and local convergence stability; after each iteration, calculate the fitness function based on the mean square error between the output torque of the BLS network and the actual frictional perturbation; the formula for calculating the shrinkage factor is: ； in: Set as the sum of individual cognitive factors and group social factors. ; Step 4.3: To improve the boundary control capability of the particle swarm optimization process, initialize the particles: x i (0)~U(x min ,x max ),v i (0)~U(v min ,v max ); Where: x i (0) represents the initial position vector of the i-th particle, v i (0) is the initial velocity vector of the i-th particle, (x min ,x max ) represents the feasible region boundary of the particle position, and represents the upper and lower bounds of the robot arm's parameter space. min ,v max This limits the range of particle velocity. The initial position and velocity of the particles are uniformly sampled from the upper and lower bounds to ensure global coverage of the search space. To prevent particles from going out of bounds and causing non-convergence, position boundary conditions are set.

5. The method for identifying and suppressing frictional vibration interference in a robotic arm based on BLS-PSO as described in claim 2, characterized in that, Step 5 specifically includes the following steps: Step 5.1: To enhance the model's fitting ability and control the accuracy of feedforward compensation, a non-structural perturbation term is introduced into the original structural model, resulting in the overall model expression as follows: τ(t)=Y(x t )P+τ f (t)+e(t); Where: τ(t) is the actual joint torque vector at the current moment, Y(x) t ) is the regression matrix, P is the structural dynamics parameter vector, and τ f e(t) represents the non-structural dynamic torque, and e(t) represents the measurement noise term. Step 5.2: By combining the nonlinear modeling properties of BLS with the global optimization of PSO, the structural model terms are estimated using the least squares method, specifically in the following form: ； The specific expression for the residual is defined as follows: ； Step 5.3: Construct the disturbance modeling function using BLS, specifically in the following form: ； in: Let F be the estimate of the unstructured disturbance at time t. BLS This represents the mapping function of BLS, where β is the connection weight of the BLS re-output layer; Step 5.4: Combine the disturbance modeling function constructed by BLS with the residual definition to finally obtain the system identification model, which is expressed in the following form: ； in: This is the final estimated total output torque. The torque is estimated based on the dynamic model of the robotic arm mechanism.

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