An industrial robot arm residual vibration suppression method based on disturbance observer and partial feature structure configuration

Abstract

This invention relates to the field of industrial robot vibration control technology, specifically to a method for suppressing residual vibration in industrial robotic arms based on a disturbance observer and partial feature structure configuration. The method first reads the robotic arm's structural parameters, working posture, and control setting parameters to establish a rigid-flexible coupling dynamic model; it then constructs a linear nominal model and a lumped disturbance model, representing inertia mismatch, damping mismatch, nonlinear friction, multi-axis coupling, and external impacts as equivalent lumped disturbances; an inner-loop disturbance observer is designed for disturbance estimation and feedforward compensation; an outer-loop partial feature structure configuration controller is established to configure the closed-loop poles and eigenvectors of the dominant flexible mode; and the control gain is updated in real-time by combining posture gain scheduling, ultimately generating joint control inputs to achieve end-effector residual vibration suppression, improving trajectory tracking performance and system robustness.

Description

Technical Field

[0001] This invention relates to the field of industrial robot dynamics modeling and active vibration suppression technology, specifically to a method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and a configuration of some characteristic structures. It falls under the categories of intelligent manufacturing, industrial robot control, rigid-flexible coupling system modeling, disturbance compensation, modal control, and high-precision motion control technology. Background Technology

[0002] Industrial robotic arms are widely used in aerospace, automotive manufacturing, electronic assembly, welding and handling, and precision machining. As production cycles increase, robotic arms need to complete high-speed positioning, rapid start-up and shutdown, and complex trajectory tracking in shorter timeframes. To reduce energy consumption, improve load-to-weight ratio, and expand workspace, robotic arm structures are increasingly moving towards lightweight, compact, and flexible designs. However, this also leads to more pronounced rigid-flexible coupling effects, low-damping structural modes, and pose-related mode drift.

[0003] During high-speed motion, the end effector of a robotic arm is susceptible to factors such as inertia mismatch, joint flexibility, nonlinear friction, multi-axis coupling, external impacts, and unmodeled high-order flexible modes, resulting in persistent residual vibrations at the end. These residual vibrations not only prolong positioning stabilization time and reduce trajectory tracking accuracy, but may also cause dynamic errors in tasks such as precision assembly, machining, and visual inspection, affecting system stability and production efficiency.

[0004] Existing methods for suppressing robotic arm vibration mainly include input shaping, PID feedback, pole placement, linear quadratic regulation, H∞ control, disturbance observer control, and active vibration suppression methods based on modal control. Among these, input shaping is simple to implement and computationally inexpensive, but it is prone to failure when modal frequency drift is caused by changes in the robotic arm's pose. Traditional PID control struggles to provide selective damping for flexible modes. Disturbance observers can estimate equivalent disturbances and perform feedforward compensation, but they are limited by Q-filter bandwidth, measurement noise, and unmodeled high-frequency dynamic constraints, making it difficult to directly suppress narrowband structural resonance. Some feature configurations can alter modal responses through closed-loop pole and eigenvector configurations, but if applied directly to uncertain rigid-flexible coupled objects, they are susceptible to nonlinear disturbances and pose-related modal drift.

[0005] Therefore, there is an urgent need for a residual vibration suppression method for industrial robotic arms that can simultaneously consider disturbance compensation, flexible modal selective damping, and pose-related modal drift, so that the robotic arm can still achieve rapid vibration attenuation, stable trajectory tracking, and robust motion control in complex workspaces without relying on direct measurement of high-frequency flexible states. Summary of the Invention

[0006] This invention aims to address the problems of slow attenuation of residual vibration at the end effector, modal frequency drift with pose, incompatibility of traditional disturbance observers in achieving both low-frequency disturbance compensation and high-frequency robust stability, and insufficient adaptability of fixed-gain modal control in existing industrial robotic arms during high-speed movement, frequent starts and stops, and complex trajectory tracking. It proposes a method for suppressing residual vibration in industrial robotic arms based on the configuration of a disturbance observer and certain feature structures. This method achieves unified suppression of disturbance-induced motion errors, flexible modal residual vibration, and pose-related modal drift in robotic arms through the coordinated design of rigid-flexible coupling dynamic modeling, linear nominal model construction, inner-loop disturbance observation compensation, outer-loop feature structure configuration, and pose-related gain scheduling.

[0007] To achieve the above objectives, the technical solution provided by this invention is as follows:

[0008] Step 1: For the target industrial robotic arm, collect data on the joint pose, joint velocity, joint acceleration, driving torque and end-effector vibration response. Combine this with the link mass distribution, joint equivalent stiffness, joint equivalent damping and reducer transmission parameters to establish a rigid-flexible coupling dynamic model of the motor side and the link side.

[0009] Step 2: Based on the rigid-flexible coupling dynamics model, construct a linear nominal model within the preset workspace, load range, and scheduling pose range, and express the difference between the actual nonlinear dynamics and the linear nominal model as an equivalent lumped disturbance;

[0010] Step 3: Design an inner-loop disturbance observer based on the linear nominal model, and obtain the equivalent lumped disturbance estimate through the nominal inverse model, the actual measured response, and a low-pass filter;

[0011] Step 4: The equivalent lumped disturbance estimate is fed back to the control input in a feedforward manner to perform dynamic compensation in the low-frequency and mid-frequency ranges for inertia mismatch, damping mismatch, nonlinear friction, multi-axis coupling and external shocks;

[0012] Step 5: Establish a state-space model based on the linear nominal model after disturbance observer compensation, and extract the natural frequencies, damping ratios, and mode shapes of the dominant flexible modes of the robotic arm through modal analysis;

[0013] Step 6: Design a partial feature structure configuration controller to selectively configure the closed-loop poles and closed-loop eigenvectors of the dominant flexible mode in order to improve the damping of the flexible mode and reduce its energy distribution in the end-vibration output.

[0014] Step 7: Based on the modal frequency drift characteristics under different joint poses, establish the scheduling relationship between pose variables and partial feature structure configuration feedback gain, and update the feedback gain in real time;

[0015] Step 8: Integrate the basic servo control quantity, the disturbance observer feedforward compensation quantity, and the modal damping control quantity of some feature structures to generate the final joint drive control input, thereby achieving residual vibration suppression of the industrial robotic arm.

[0016] Furthermore, the establishment of the rigid-flexible coupling dynamic model in step 1 includes the following steps:

[0017] Step 1.1: Establish the generalized coordinate q on the motor side of the robotic arm m Generalized coordinates q on the connecting rod side l The corresponding velocities and accelerations are used, and the mapping relationship between the angular velocities and linear velocities of each link of the robotic arm and the joint motion variables is described by the Jacobian matrix.

[0018] Step 1.2: Based on the principles of Euler-Lagrange dynamics, establish the system's kinetic energy T, potential energy U, and dissipation function D, respectively, in the following specific forms:

[0019]

[0020]

[0021] in, and Let these represent the angular displacement vectors on the rotor side and the connecting rod side of the motor, respectively. This is a diagonal matrix that includes the inertia of the motor rotor and the input of the reducer. The inertial matrix is ​​a symmetric positive definite matrix related to the linkage configuration; to quantify the non-conserved energy within the system, a depletion function is introduced. Let B be the rate at which kinetic energy is converted during the system's motion. v This is the joint equivalent viscous damping matrix;

[0022] Step 1.3: Substitute the kinetic energy, potential energy, and dissipation function into the generalized Euler-Lagrange equations to obtain the rigid-flexible coupling dynamic equations for the motor side and the connecting rod side, specifically in the following form:

[0023]

[0024] in This refers to the nonlinear frictional torque at the motor end. The external disturbance torque at the connecting rod end; This refers to the electromagnetic driving torque of the motor.

[0025] Step 1.4: The rigid-flexible coupling dynamic equation is used to characterize the structural vibration response excited after the motor output torque is transmitted to the connecting rod side through the flexible joint, and provides a dynamic basis for the subsequent establishment of the linear nominal model, disturbance observer and partial characteristic structure configuration controller.

[0026] Furthermore, the establishment of the linear nominal model and the equivalent lumped disturbance model in step 2 includes the following steps:

[0027] Step 2.1: Within the preset workspace, load range, and scheduling pose range, the flexible deformation of the robotic arm is regarded as a small deformation, and the elastic restoring torque and damping torque are locally expressed as the nominal stiffness matrix K and the nominal damping matrix Bv;

[0028] Step 2.2: Based on the CAD mass distribution of the robotic arm, the inertia parameters of the connecting rods and motors, the results of finite element modal analysis, and the modal damping ratios identified experimentally, determine the nominal inertia matrix, nominal damping matrix, and nominal stiffness matrix, and construct a local linear nominal model, specifically in the form of:

[0029]

[0030]

[0031]

[0032] Where x is the state variable, A is the nominal state matrix, B is the nominal input matrix, C is the output matrix, and τ link To control the input torque;

[0033] Step 2.3: The part of the actual robotic arm's nonlinear dynamics not included in the linear nominal model is uniformly represented as an equivalent lumped disturbance, the specific form of which is:

[0034]

[0035] The first term is for inertial mismatch; the second term is for damping mismatch; the third term is for nonlinear frictional torque; the fourth term is for multiaxial dynamic coupling; and the fifth term is for external impact.

[0036] Step 2.4: Map the disturbance on the connecting rod side to the torque channel on the motor side through the flexible joint transmission relationship, so that the equivalent lumped disturbance is represented as a matched disturbance in the same channel as the control input, thereby satisfying the estimation and feedforward compensation conditions of the disturbance observer.

[0037] Furthermore, the design and compensation process of the inner loop disturbance observer described in steps 3 and 4 includes the following steps:

[0038] Step 3.1: Using the linear nominal model P n (s) serves as the reference model for the disturbance observer and is reconstructed through the nominal inverse model to generate the equivalent input required for the actual measurement response;

[0039] Step 3.2: The difference between the equivalent input and the actual control input is used as the initial disturbance estimation signal, and after processing by a low-pass filter Q(s), the lumped disturbance estimation value is obtained, specifically in the form of:

[0040]

[0041] Step 3.3: Construct a disturbance compensation control quantity based on the disturbance estimate, and superimpose it onto the basic servo control quantity in a feedforward manner. Specifically, the form is as follows:

[0042]

[0043] Step 3.4: The disturbance observer mainly compensates for low-frequency and mid-frequency disturbance components within the effective bandwidth, and retains high-frequency noise that cannot be fully compensated, unmodeled high-order flexible modes, and rapidly changing disturbances as bounded residual uncertainties.

[0044] Step 3.5: Establish the actual controlled object P(s) and the nominal model P n(s) The multiplicative uncertainty relationship between them is specifically in the form of:

[0045]

[0046] Where Δ(s) represents the multiplicative uncertainty of the actual object relative to the nominal model;

[0047] Step 3.6: Based on the structural modal frequencies of the robotic arm, the measurement noise frequency band, and the unmodeled high-frequency flexible dynamics, determine the upper bound of the uncertainty, and ensure that the low-pass filter satisfies the following robust stability constraints:

[0048]

[0049]

[0050] Where ω is the frequency variable; the above constraints prevent the perturbation observer bandwidth from blindly extending to the structural resonance frequency band, and avoid amplification of high-frequency measurement noise and unmodeled modes.

[0051] Furthermore, the modal analysis and partial feature structure configuration controller establishment described in steps 5 and 6 include the following steps:

[0052] Step 5.1: Express the linear nominal model after the disturbance observer compensation in state space, and extract the rigid motion state and flexible vibration state of the robotic arm to form closed-loop state variables for the configuration of some feature structures;

[0053] Step 5.2: Analyze the modal characteristics of the robotic arm in different poses, extract the natural frequency, damping ratio and mode shape of the dominant flexible mode, and determine the target flexible mode that contributes significantly to the residual vibration at the end.

[0054] Step 5.2.1: Perform detrending processing and bandpass filtering on the vibration signal x at the end of the robotic arm to remove rigid body motion components and measurement noise;

[0055] Step 5.2.2: Construct an analytic signal using the Hilbert transform, specifically in the following form:

[0056]

[0057] Where H[x(t)] is the Hilbert transform of x, A(t) is the magnitude envelope, and φ(t) is the instantaneous phase;

[0058] Step 5.2.3: Obtain the instantaneous frequency based on the instantaneous phase, specifically in the following form:

[0059]

[0060] Step 5.2.4: During the vibration decay stage, estimate the modal damping ratio based on the logarithmic decay law of the amplitude envelope, specifically in the following form:

[0061]

[0062] Where ζ is the modal damping ratio; for multimodal coupling conditions, wavelet transform is used to extract the frequency and amplitude features corresponding to each modal ridge, and the extraction results are used as the basis for setting the parameters of the disturbance observer filter and scheduling the feedback gain;

[0063] Step 5.3: The natural frequencies and mode shapes satisfy the following generalized eigenvalue equations:

[0064]

[0065]

[0066] Where K(q) and M(q) are the pose-related stiffness matrix and mass matrix, respectively, ωi is the i-th natural frequency, and Φi is the corresponding mode shape;

[0067] Step 6.1: For the dominant flexible mode that contributes significantly to the end-effector vibration response, set a target closed-loop pole to shift the flexible mode pole to the high-damping region. The specific form of the target pole is as follows:

[0068]

[0069] Step 6.2: Establish the constraint relationship between closed-loop poles and closed-loop eigenvectors, specifically in the form of:

[0070]

[0071] Step 6.3: While configuring the flexible modal poles and eigenvectors, retain the rigid modal feature structure related to rigid body trajectory tracking, so that the trajectory tracking performance of the robotic arm is not significantly weakened by the enhancement of flexible modal damping.

[0072] Furthermore, the pose-related gain scheduling mechanism described in step 7 includes the following steps:

[0073] Step 7.1: Select joint variables that are sensitive to the frequency changes of the dominant flexible mode of the robotic arm as scheduling variables, and calculate the corresponding partial feature structure configuration feedback gain matrix offline under several typical joint poses;

[0074] Step 7.2: Establish the interpolation scheduling relationship between the current joint pose q and the feedback gain matrix, specifically in the form of:

[0075]

[0076] Step 7.3: When the robotic arm pose changes, calculate the scheduling weights in real time based on the current joint pose and update K(q). s This ensures that the dominant flexible mode maintains the target closed-loop damping characteristics under different poses;

[0077] Step 7.4: The pose-related gain scheduling mechanism and the inner-loop disturbance observer share the same linear nominal model and residual uncertainty constraints, thereby achieving coordinated control of disturbance compensation, flexible mode damping enhancement and pose-related mode drift compensation.

[0078] Furthermore, the fusion process of the final joint drive control input described in step 8 includes the following steps:

[0079] Step 8.1: Obtain the trajectory tracking control quantity output by the basic servo controller to ensure that the joint position, speed and acceleration of the robotic arm track the preset trajectory;

[0080] Step 8.2: Obtain the lumped disturbance estimate d output by the disturbance observer, and invert it to use as the feedforward compensation control quantity;

[0081] Step 8.3: Obtain the partial feature structure configuration feedback gain after pose scheduling, and generate modal damping control quantity based on the current state variable x;

[0082] Step 8.4: Fuse the basic servo control input, disturbance feedforward compensation control input, and modal damping control input to obtain the final joint drive control input, specifically in the form of:

[0083]

[0084] Step 8.5: Send the final joint drive control input to the robotic arm servo driver to enable the robotic arm to reduce the residual vibration amplitude at the end of the arm, shorten the settling time, reduce the overshoot, and enhance the robustness in high-speed positioning, frequent start-stop, impact response, and complex trajectory tracking tasks.

[0085] The residual vibration suppression method for industrial robotic arms proposed in this invention, based on a disturbance observer and a configuration of certain characteristic structures, has the following advantages compared with existing technologies:

[0086] 1. Good residual vibration suppression effect: Selective damping enhancement of the dominant flexible mode is achieved by using the outer ring PESA, which causes the poles of the flexible mode to migrate to the high damping region, thereby shortening the end vibration decay time and reducing the residual vibration amplitude after high-speed start-stop.

[0087] 2. Strong anti-disturbance capability: By estimating and feedforward compensating for inertia mismatch, damping mismatch, nonlinear friction, multi-axis coupling and external shocks through inner loop DOB, the actual controlled object is close to the linear nominal model within the effective bandwidth, thereby improving the robustness of the control system.

[0088] 3. Strong adaptability to pose changes: By updating the PESA feedback gain in real time through pose-related gain scheduling, it can compensate for the modal frequency drift caused by changes in the robot arm configuration and avoid performance degradation when the fixed gain is controlled throughout the entire workspace.

[0089] 4. Direct measurement without relying on high-frequency flexible state: This method achieves flexible mode suppression through the coordinated use of nominal model, disturbance observer and modal parameter scheduling, which can reduce the dependence on high-frequency sensors and complex multi-sensor fusion systems.

[0090] 5. Good adaptability to industrial applications: This method can be embedded in existing servo control structures and work in conjunction with basic position, speed or torque controllers. It is suitable for six-degree-of-freedom industrial robotic arms, lightweight rigid-flexible coupling robotic arms and multi-degree-of-freedom electromechanical systems with flexible joint transmissions. Attached Figure Description

[0091] Figure 1 This is a flowchart of the residual vibration suppression method for industrial robotic arms based on a disturbance observer and the configuration of some feature structures proposed in this invention.

[0092] Figure 2 This is a system architecture diagram of the enhanced disturbance observer in this invention, used to illustrate the connection relationship between the nominal model, Q filter, disturbance estimation and feedforward compensation.

[0093] Figure 3This is a block diagram of the DOB-PESA composite control system in this invention, used to illustrate the cooperative control relationship between basic servo control, disturbance observation compensation, partial feature structure configuration control, and pose gain scheduling.

[0094] Figure 4 The figure shows the deformation and modal analysis results of the first dominant flexible mode of the robotic arm.

[0095] Figure 5 The figure shows the deformation and modal analysis results of the second-order dominant flexible mode of the robotic arm.

[0096] Figure 6 The figure shows the deformation and modal analysis results of the third dominant flexible mode of the robotic arm.

[0097] Figure 7 The figure shows the deformation and modal analysis results of the fourth dominant flexible mode of the robotic arm.

[0098] Figure 8 The figure shows the deformation and modal analysis results of the fifth dominant flexible mode of the robotic arm.

[0099] Figure 9 The figure shows the deformation and modal analysis results of the sixth dominant flexible mode of the robotic arm.

[0100] Figure 10 The step response curves of joint 1 of the robotic arm under different control algorithms are shown.

[0101] Figure 11 The step response curves of joint 2 of the robotic arm under different control algorithms are shown.

[0102] Figure 12 The residual vibration suppression performance curves of joint 1 of the robotic arm are shown under different control algorithms.

[0103] Figure 13 The residual vibration suppression performance curves of joint 2 of the robotic arm under different control algorithms are shown.

[0104] Figure 14 This is a diagram showing the vibration measurement and control effect of channel 15 under the complex trajectory operation conditions of a real robotic arm.

[0105] Figure 15 This is a diagram showing the vibration measurement and control effect of channel 16 under the complex trajectory operation conditions of a real robotic arm. Specific implementation methods

[0106] 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 scope of protection of the present application.

[0107] Example 1

[0108] This embodiment uses a six-degree-of-freedom industrial robotic arm as an example to verify the residual vibration suppression method based on disturbance observers and partial feature structure configuration proposed in this invention. The experimental system includes the six-degree-of-freedom industrial robotic arm body, a servo drive system, an end-effector vibration measurement module, a data acquisition module, and a host computer control and analysis platform. During the experiment, the robotic arm performs step response, pulse disturbance response, and complex trajectory tracking tasks, collecting joint pose, velocity, drive torque, and end-effector vibration response data. Vibration signals are collected by multi-channel sensors deployed at the end of the robotic arm and related structural locations, and recorded by the data acquisition system for analyzing the residual vibration attenuation effect under different control strategies. The process includes the following steps:

[0109] Step 1: For the target industrial robotic arm, collect data on the joint pose, joint velocity, joint acceleration, driving torque and end-effector vibration response. Combine this with the link mass distribution, joint equivalent stiffness, joint equivalent damping and reducer transmission parameters to establish a rigid-flexible coupling dynamic model of the motor side and the link side.

[0110] Step 2: Based on the rigid-flexible coupling dynamics model, construct a linear nominal model within the preset workspace, load range, and scheduling pose range, and express the difference between the actual nonlinear dynamics and the linear nominal model as an equivalent lumped disturbance;

[0111] Step 3: Design an inner-loop disturbance observer based on the linear nominal model, and obtain the equivalent lumped disturbance estimate through the nominal inverse model, the actual measured response, and a low-pass filter;

[0112] Step 4: The equivalent lumped disturbance estimate is fed back to the control input in a feedforward manner to perform dynamic compensation in the low-frequency and mid-frequency ranges for inertia mismatch, damping mismatch, nonlinear friction, multi-axis coupling and external shocks;

[0113] Step 5: Establish a state-space model based on the linear nominal model after disturbance observer compensation, and extract the natural frequencies, damping ratios, and mode shapes of the dominant flexible modes of the robotic arm through modal analysis;

[0114] Step 6: Design a partial feature structure configuration controller to selectively configure the closed-loop poles and closed-loop eigenvectors of the dominant flexible mode in order to improve the damping of the flexible mode and reduce its energy distribution in the end-vibration output.

[0115] Step 7: Based on the modal frequency drift characteristics under different joint poses, establish the scheduling relationship between pose variables and partial feature structure configuration feedback gain, and update the feedback gain in real time;

[0116] Step 8: Integrate the basic servo control quantity, the disturbance observer feedforward compensation quantity, and the modal damping control quantity of some feature structures to generate the final joint drive control input, thereby achieving residual vibration suppression of the industrial robotic arm.

[0117] Furthermore, the establishment of the rigid-flexible coupling dynamic model in step 1 includes the following steps:

[0118] Step 1.1: Establish the generalized coordinate q on the motor side of the robotic arm m Generalized coordinates q on the connecting rod side l The corresponding velocities and accelerations are used, and the mapping relationship between the angular velocities and linear velocities of each link of the robotic arm and the joint motion variables is described by the Jacobian matrix.

[0119] Step 1.2: Based on the principles of Euler-Lagrange dynamics, establish the system's kinetic energy T, potential energy U, and dissipation function D, respectively, in the following specific forms:

[0120]

[0121]

[0122] in, and Let these represent the angular displacement vectors on the rotor side and the connecting rod side of the motor, respectively. This is a diagonal matrix that includes the inertia of the motor rotor and the input of the reducer. The inertial matrix is ​​a symmetric positive definite matrix related to the linkage configuration; to quantify the non-conserved energy within the system, a depletion function is introduced. Let B be the rate at which kinetic energy is converted during the system's motion. v This is the joint equivalent viscous damping matrix;

[0123] Step 1.3: Substitute the kinetic energy, potential energy, and dissipation function into the generalized Euler-Lagrange equations to obtain the rigid-flexible coupling dynamic equations for the motor side and the connecting rod side, specifically in the following form:

[0124]

[0125] in This refers to the nonlinear frictional torque at the motor end. The external disturbance torque at the connecting rod end; This refers to the electromagnetic driving torque of the motor.

[0126] Step 1.4: The rigid-flexible coupling dynamic equation is used to characterize the structural vibration response excited after the motor output torque is transmitted to the connecting rod side through the flexible joint, and provides a dynamic basis for the subsequent establishment of the linear nominal model, disturbance observer and partial characteristic structure configuration controller.

[0127] Furthermore, the establishment of the linear nominal model and the equivalent lumped disturbance model in step 2 includes the following steps:

[0128] Step 2.1: Within the preset workspace, load range, and scheduling pose range, the flexible deformation of the robotic arm is regarded as a small deformation, and the elastic restoring torque and damping torque are locally expressed as the nominal stiffness matrix K and the nominal damping matrix Bv;

[0129] Step 2.2: Based on the CAD mass distribution of the robotic arm, the inertia parameters of the connecting rods and motors, the results of finite element modal analysis, and the modal damping ratios identified experimentally, determine the nominal inertia matrix, nominal damping matrix, and nominal stiffness matrix, and construct a local linear nominal model, specifically in the form of:

[0130]

[0131]

[0132]

[0133] Where x is the state variable, A is the nominal state matrix, B is the nominal input matrix, C is the output matrix, and τ link To control the input torque;

[0134] Step 2.3: The part of the actual robotic arm's nonlinear dynamics not included in the linear nominal model is uniformly represented as an equivalent lumped disturbance, the specific form of which is:

[0135]

[0136] The first term is for inertial mismatch; the second term is for damping mismatch; the third term is for nonlinear frictional torque; the fourth term is for multiaxial dynamic coupling; and the fifth term is for external impact.

[0137] Step 2.4: Map the disturbance on the connecting rod side to the torque channel on the motor side through the flexible joint transmission relationship, so that the equivalent lumped disturbance is represented as a matched disturbance in the same channel as the control input, thereby satisfying the estimation and feedforward compensation conditions of the disturbance observer.

[0138] Furthermore, the design and compensation process of the inner loop disturbance observer described in steps 3 and 4 includes the following steps:

[0139] Step 3.1: Using the linear nominal model P n (s) serves as the reference model for the disturbance observer and is reconstructed through the nominal inverse model to generate the equivalent input required for the actual measurement response;

[0140] Step 3.2: The difference between the equivalent input and the actual control input is used as the initial disturbance estimation signal, and after processing by a low-pass filter Q(s), the lumped disturbance estimation value is obtained, specifically in the form of:

[0141]

[0142] Step 3.3: Construct a disturbance compensation control quantity based on the disturbance estimate, and superimpose it onto the basic servo control quantity in a feedforward manner. Specifically, the form is as follows:

[0143]

[0144] Step 3.4: The disturbance observer mainly compensates for low-frequency and mid-frequency disturbance components within the effective bandwidth, and retains high-frequency noise that cannot be fully compensated, unmodeled high-order flexible modes, and rapidly changing disturbances as bounded residual uncertainties.

[0145] Step 3.5: Establish the actual controlled object P(s) and the nominal model P n(s) The multiplicative uncertainty relationship between them is specifically in the form of:

[0146]

[0147] Where Δ(s) represents the multiplicative uncertainty of the actual object relative to the nominal model;

[0148] Step 3.6: Based on the structural modal frequencies of the robotic arm, the measurement noise frequency band, and the unmodeled high-frequency flexible dynamics, determine the upper bound of the uncertainty, and ensure that the low-pass filter satisfies the following robust stability constraints:

[0149]

[0150]

[0151] Where ω is the frequency variable; the above constraints prevent the perturbation observer bandwidth from blindly extending to the structural resonance frequency band, and avoid amplification of high-frequency measurement noise and unmodeled modes.

[0152] Furthermore, the modal analysis and partial feature structure configuration controller establishment described in steps 5 and 6 include the following steps:

[0153] Step 5.1: Express the linear nominal model after the disturbance observer compensation in state space, and extract the rigid motion state and flexible vibration state of the robotic arm to form closed-loop state variables for the configuration of some feature structures;

[0154] Step 5.2: Analyze the modal characteristics of the robotic arm in different poses, extract the natural frequency, damping ratio and mode shape of the dominant flexible mode, and determine the target flexible mode that contributes significantly to the residual vibration at the end.

[0155] Step 5.2.1: Perform detrending processing and bandpass filtering on the vibration signal x at the end of the robotic arm to remove rigid body motion components and measurement noise;

[0156] Step 5.2.2: Construct an analytic signal using the Hilbert transform, specifically in the following form:

[0157]

[0158] Where H[x(t)] is the Hilbert transform of x, A(t) is the magnitude envelope, and φ(t) is the instantaneous phase;

[0159] Step 5.2.3: Obtain the instantaneous frequency based on the instantaneous phase, specifically in the following form:

[0160]

[0161] Step 5.2.4: During the vibration decay stage, estimate the modal damping ratio based on the logarithmic decay law of the amplitude envelope, specifically in the following form:

[0162]

[0163] Where ζ is the modal damping ratio; for multimodal coupling conditions, wavelet transform is used to extract the frequency and amplitude features corresponding to each modal ridge, and the extraction results are used as the basis for setting the parameters of the disturbance observer filter and scheduling the feedback gain;

[0164] Step 5.3: The natural frequencies and mode shapes satisfy the following generalized eigenvalue equations:

[0165]

[0166]

[0167] Where K(q) and M(q) are the pose-related stiffness matrix and mass matrix, respectively, ωi is the i-th natural frequency, and Φi is the corresponding mode shape;

[0168] Step 6.1: For the dominant flexible mode that contributes significantly to the end-effector vibration response, set a target closed-loop pole to shift the flexible mode pole to the high-damping region. The specific form of the target pole is as follows:

[0169]

[0170] Step 6.2: Establish the constraint relationship between closed-loop poles and closed-loop eigenvectors, specifically in the form of:

[0171]

[0172] Step 6.3: While configuring the flexible modal poles and eigenvectors, retain the rigid modal feature structure related to rigid body trajectory tracking, so that the trajectory tracking performance of the robotic arm is not significantly weakened by the enhancement of flexible modal damping.

[0173] Furthermore, the pose-related gain scheduling mechanism described in step 7 includes the following steps:

[0174] Step 7.1: Select joint variables that are sensitive to the frequency changes of the dominant flexible mode of the robotic arm as scheduling variables, and calculate the corresponding partial feature structure configuration feedback gain matrix offline under several typical joint poses;

[0175] Step 7.2: Establish the interpolation scheduling relationship between the current joint pose q and the feedback gain matrix, specifically in the form of:

[0176]

[0177] Step 7.3: When the robotic arm pose changes, calculate the scheduling weights in real time based on the current joint pose and update K(q). s This ensures that the dominant flexible mode maintains the target closed-loop damping characteristics under different poses;

[0178] Step 7.4: The pose-related gain scheduling mechanism and the inner-loop disturbance observer share the same linear nominal model and residual uncertainty constraints, thereby achieving coordinated control of disturbance compensation, flexible mode damping enhancement and pose-related mode drift compensation.

[0179] Furthermore, the fusion process of the final joint drive control input described in step 8 includes the following steps:

[0180] Step 8.1: Obtain the trajectory tracking control quantity output by the basic servo controller to ensure that the joint position, speed and acceleration of the robotic arm track the preset trajectory;

[0181] Step 8.2: Obtain the lumped disturbance estimate d output by the disturbance observer, and invert it to use as the feedforward compensation control quantity;

[0182] Step 8.3: Obtain the partial feature structure configuration feedback gain after pose scheduling, and generate modal damping control quantity based on the current state variable x;

[0183] Step 8.4: Fuse the basic servo control input, disturbance feedforward compensation control input, and modal damping control input to obtain the final joint drive control input, specifically in the form of:

[0184]

[0185] Step 8.5: Send the final joint drive control input to the robotic arm servo driver to enable the robotic arm to reduce the residual vibration amplitude at the end of the arm, shorten the settling time, reduce the overshoot, and enhance the robustness in high-speed positioning, frequent start-stop, impact response, and complex trajectory tracking tasks.

[0186] To verify the effectiveness of the method of this invention, simulation experiments and real robotic arm experiments were conducted in this embodiment. The simulation experiment was based on the Aubo i5 six-degree-of-freedom industrial robotic arm rigid-flexible coupling dynamic model, with a maximum load of 10 kg. In the modal analysis, the main body material of the robotic arm was set as 6061 aluminum alloy, and a 10 kg vertical load was applied at the end. The modal analysis results showed that the first and second dominant flexible mode frequencies of the robotic arm were 41.837 Hz and 42.029 Hz, respectively. To verify the effect of pose-related gain scheduling, 280 Hz, 300 Hz, and 320 Hz were selected as representative frequency points under different stiffness conditions, with 300 Hz as the nominal design frequency and the target damping ratio set to 0.707. Regarding control parameters, the traditional PID controller parameters are set as follows: Kp=[420,450,380,240,160,120], Ki=[35,38,30,20,12,10], Kd=[30,32,27,18,12,9]; the disturbance observer uses a low-pass filter Q(s)=1 / (0.005s+1)², the filter time constant is 0.005s, and the approximate cutoff frequency is 31.83Hz; the expected closed-loop poles of the dominant flexible mode corresponding to the PESA controller are −185.85±185.90j and −186.70±186.76j.

[0187] The real-world robotic arm experiment used the Aubo i5 six-DOF industrial robotic arm platform as the subject. The experimental system included the robotic arm body, servo drive system, end-effector vibration measurement module, 16-channel data acquisition module, and host computer control and analysis platform. During the experiment, the robotic arm executed a complex trajectory for 13 seconds, and multi-channel acceleration response signals were acquired from the end-effector and related structural positions. Comparison strategies included no control, traditional PID control, standard disturbance observer control, single PESA control, and the DOB-PESA composite control method of this invention. Evaluation indicators included joint step response overshoot, settling time, end-effector residual vibration amplitude, vibration decay rate, and multi-channel vibration response under complex trajectories.

[0188] like Figure 1As shown, the method of this invention first reads the structural parameters, working posture, and control setting parameters of the robotic arm, then establishes a rigid-flexible coupling dynamic model, and further constructs a linear nominal model and a lumped disturbance model. Based on this, an inner-loop disturbance observer is used to estimate and compensate for low-frequency and mid-frequency disturbances, an outer-loop partial feature structure configuration controller is used to selectively dampen and enhance the dominant flexible mode, and pose-related gain scheduling is used to compensate for modal frequency drift caused by changes in the robotic arm configuration. Finally, the DOB compensation and PESA control quantities are fused to generate joint control inputs, achieving end-effector residual vibration suppression.

[0189] like Figure 2 As shown, the enhanced disturbance observer is based on a linear nominal model. It estimates the deviation between the actual output and the nominal output using the nominal inverse model and a Q-filter, obtaining an equivalent lumped disturbance. This lumped disturbance includes low-frequency or mid-frequency disturbance components such as inertia mismatch, damping mismatch, nonlinear friction, multi-axis coupling, and external shocks. The estimated disturbance is compensated to the control input via feedforward, ensuring that the actual controlled object of the robotic arm approximates the linear nominal model within the effective bandwidth of the disturbance observer.

[0190] like Figure 3 As shown, the DOB-PESA composite control system consists of a basic servo controller, an inner-loop disturbance observer, an outer-loop partial feature structure configuration controller, and a pose-related gain scheduling module. The basic servo controller ensures joint trajectory tracking, the disturbance observer compensates for equivalent lumped disturbances, the partial feature structure configuration controller improves the damping of the dominant flexible modes, and the gain scheduling module adjusts the PESA feedback gain in real time according to the current joint pose. Through the synergistic effect of these modules, disturbance-induced errors, structural flexible vibrations, and pose-related modal drift can be suppressed simultaneously.

[0191] like Figures 4 to 9 As shown, modal analysis of the robotic arm yields the deformation characteristics, natural frequencies, and mode shapes of its dominant flexible modes. This figure illustrates that the residual vibration at the robotic arm's end effector is not solely caused by rigid body motion errors, but is closely related to the structural flexible modes. Therefore, this invention does not merely suppress errors by increasing servo feedback gain, but further selects the flexible modes that contribute significantly to the residual vibration at the end effector as the target modes, and selectively dampes and enhances them through a partial feature structure configuration method, thereby increasing the vibration decay rate.

[0192] like Figures 10 to 11As shown in the joint step response comparison experiment, although traditional PID control can achieve basic position tracking, it still exhibits significant overshoot and a long settling time under high-speed step input. Standard disturbance observer control can compensate for some low-frequency disturbances, but its effect on attenuating flexible modal vibrations is limited. PESA control alone can improve flexible modal damping, but its response stability is affected by model mismatch and external disturbances. In contrast, the DOB-PESA composite control method of this invention, through the synergistic effect of inner-loop disturbance compensation and outer-loop modal damping enhancement, can reduce step response overshoot, shorten settling time, and improve joint dynamic response stability.

[0193] like Figures 12 to 13 As shown in the comparative experiment on residual vibration suppression of the end effector, under no control or traditional PID control conditions, the vibration at the end of the robotic arm decays slowly, and the residual vibration lasts for a long time. Standard DOB control mainly improves low-frequency motion errors caused by disturbances, but cannot directly change the closed-loop poles of the flexible mode. PESA control alone can improve the damping of the flexible mode, but it is more sensitive to nonlinear disturbances and pose changes. The method of this invention approximates the controlled object to a linear nominal model within the effective bandwidth using DOB, and then uses PESA to configure the dominant flexible mode with high damping, combined with pose gain scheduling to compensate for mode frequency drift. Therefore, it can more significantly reduce the amplitude of residual vibration at the end and accelerate the vibration decay process.

[0194] like Figures 14 to 15 As shown, the actual control effect of the method of the present invention is verified by multi-channel vibration measurement data under the complex trajectory operation conditions of a real robotic arm. In the experiment, the robotic arm performs continuous trajectory operation, and each measurement channel records the vibration response of the end effector and related structural positions. The comparative results show that the method of the present invention can still maintain a good vibration suppression effect under complex trajectory, pose change and multi-channel vibration coupling conditions, indicating that the method is not only applicable to single step response or simulated working conditions, but also applicable to complex operation scenarios of real industrial robotic arms.

Claims

1. A method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and a configuration of certain characteristic structures, characterized in that, The method includes: Step 1: For the target industrial robotic arm, collect data on the joint pose, joint velocity, joint acceleration, driving torque and end-effector vibration response. Combine this with the link mass distribution, joint equivalent stiffness, joint equivalent damping and reducer transmission parameters to establish a rigid-flexible coupling dynamic model of the motor side and the link side. Step 2: Based on the rigid-flexible coupling dynamics model, construct a linear nominal model within the preset workspace, load range, and scheduling pose range, and express the difference between the actual nonlinear dynamics and the linear nominal model as an equivalent lumped disturbance; Step 3: Design an inner-loop disturbance observer based on the linear nominal model, and obtain the equivalent lumped disturbance estimate through the nominal inverse model, the actual measured response, and a low-pass filter; Step 4: The equivalent lumped disturbance estimate is fed back to the control input in a feedforward manner to perform dynamic compensation in the low-frequency and mid-frequency ranges for inertia mismatch, damping mismatch, nonlinear friction, multi-axis coupling and external shocks; Step 5: Establish a state-space model based on the linear nominal model after disturbance observer compensation, and extract the natural frequencies, damping ratios, and mode shapes of the dominant flexible modes of the robotic arm through modal analysis; Step 6: Design a partial feature structure configuration controller to selectively configure the closed-loop poles and closed-loop eigenvectors of the dominant flexible mode in order to improve the damping of the flexible mode and reduce its energy distribution in the end-vibration output. Step 7: Based on the modal frequency drift characteristics under different joint poses, establish the scheduling relationship between pose variables and partial feature structure configuration feedback gain, and update the feedback gain in real time; Step 8: Integrate the basic servo control quantity, the disturbance observer feedforward compensation quantity, and the modal damping control quantity of some feature structures to generate the final joint drive control input, thereby achieving residual vibration suppression of the industrial robotic arm.

2. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 1, characterized in that, The establishment of the rigid-flexible coupling dynamic model in step 1 includes the following steps: Step 1.1: Establish the generalized coordinates q on the motor side of the robot arm m , the generalized coordinates q on the link side l , and their corresponding velocities and accelerations, and describe the mapping relationship between the angular velocity, linear velocity of each link of the robot arm and the joint motion variables through the Jacobian matrix; Step 1.2: Based on the principles of Euler-Lagrange dynamics, establish the system's kinetic energy T, potential energy U, and dissipation function D, respectively, in the following specific forms: in, and Let these represent the angular displacement vectors on the rotor side and the connecting rod side of the motor, respectively. This is a diagonal matrix that includes the inertia of the motor rotor and the input of the reducer. The inertial matrix is ​​a symmetric positive definite matrix related to the linkage configuration; to quantify the non-conserved energy within the system, a depletion function is introduced. Let B be the rate at which kinetic energy is converted during the system's motion. v This is the joint equivalent viscous damping matrix; Step 1.3: Substitute the kinetic energy, potential energy, and dissipation function into the generalized Euler-Lagrange equations to obtain the rigid-flexible coupling dynamic equations for the motor side and the connecting rod side, specifically in the following form: in, This refers to the nonlinear frictional torque at the motor end. The external disturbance torque at the connecting rod end; This refers to the electromagnetic driving torque of the motor. Step 1.4: The rigid-flexible coupling dynamic equation is used to characterize the structural vibration response excited after the motor output torque is transmitted to the connecting rod side through the flexible joint, and provides a dynamic basis for the subsequent establishment of the linear nominal model, disturbance observer and partial characteristic structure configuration controller.

3. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 1, characterized in that, Step 2 involves establishing the linear nominal model and the equivalent lumped disturbance model, which includes the following steps: Step 2.1: Within the preset workspace, load range, and scheduling pose range, the flexible deformation of the robotic arm is regarded as a small deformation, and the elastic restoring torque and damping torque are locally expressed as the nominal stiffness matrix K and the nominal damping matrix Bv; Step 2.2: Based on the CAD mass distribution of the robotic arm, the inertia parameters of the connecting rods and motors, the results of finite element modal analysis, and the modal damping ratios identified experimentally, determine the nominal inertia matrix, nominal damping matrix, and nominal stiffness matrix, and construct a local linear nominal model, specifically in the form of: where x is a state variable, A is a nominal state matrix, B is a nominal input matrix, C is an output matrix, τ link is a control input torque; Step 2.3: The part of the actual robotic arm's nonlinear dynamics not included in the linear nominal model is uniformly represented as an equivalent lumped disturbance, the specific form of which is: The first term is for inertial mismatch; the second term is for damping mismatch; the third term is for nonlinear frictional torque; the fourth term is for multiaxial dynamic coupling; and the fifth term is for external impact. Step 2.4: Map the disturbance on the connecting rod side to the torque channel on the motor side through the flexible joint transmission relationship, so that the equivalent lumped disturbance is represented as a matched disturbance in the same channel as the control input, thereby satisfying the estimation and feedforward compensation conditions of the disturbance observer.

4. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 1, characterized in that, The design and compensation process of the inner loop disturbance observer described in steps 3 and 4 includes the following steps: Step 3.1: Using the linear nominal model P n (s) serves as the reference model for the disturbance observer and is reconstructed through the nominal inverse model to generate the equivalent input required for the actual measurement response; Step 3.2: The difference between the equivalent input and the actual control input is used as the initial disturbance estimation signal, and after processing by the low-pass filter Q(s), the lumped disturbance estimation value is obtained, specifically in the form of: Step 3.3: Construct a disturbance compensation control quantity based on the disturbance estimate, and superimpose it onto the basic servo control quantity in a feedforward manner. Specifically, the form is as follows: Step 3.4: The disturbance observer mainly compensates for low-frequency and mid-frequency disturbance components within the effective bandwidth, and retains high-frequency noise that cannot be fully compensated, unmodeled high-order flexible modes, and rapidly changing disturbances as bounded residual uncertainties.

5. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 4, characterized in that, The bandwidth of the low-pass filter Q(s) is limited by multiplicative uncertainty constraints and small gain stability conditions. Its design process includes the following steps: Step 3.5: Establish the actual controlled object P(s) and the nominal model P n(s) The multiplicative uncertainty relationship between them is specifically in the form of: Where Δ(s) represents the multiplicative uncertainty of the actual object relative to the nominal model; Step 3.6: Based on the structural modal frequencies of the robotic arm, the measurement noise frequency band, and the unmodeled high-frequency flexible dynamics, determine the upper bound of the uncertainty, and ensure that the low-pass filter satisfies the following robust stability constraints: Where ω is the frequency variable; the above constraints prevent the perturbation observer bandwidth from blindly extending to the structural resonance frequency band, and avoid amplification of high-frequency measurement noise and unmodeled modes.

6. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 1, characterized in that, Steps 5 and 6, which involve modal analysis and establishing a partial feature structure configuration controller, include the following steps: Step 5.1: Express the linear nominal model after the disturbance observer compensation in state space, and extract the rigid motion state and flexible vibration state of the robotic arm to form closed-loop state variables for the configuration of some feature structures; Step 5.2: Analyze the modal characteristics of the robotic arm in different poses, extract the natural frequency, damping ratio and mode shape of the dominant flexible mode, and determine the target flexible mode that contributes significantly to the residual vibration at the end. Step 5.2.1: Perform detrending processing and bandpass filtering on the vibration signal x at the end of the robotic arm to remove rigid body motion components and measurement noise; Step 5.2.2: Construct an analytic signal using the Hilbert transform, specifically in the following form: Where H[x(t)] is the Hilbert transform of x, A(t) is the magnitude envelope, and φ(t) is the instantaneous phase; Step 5.2.3: Obtain the instantaneous frequency based on the instantaneous phase, specifically in the following form: Step 5.2.4: During the vibration decay stage, estimate the modal damping ratio based on the logarithmic decay law of the amplitude envelope, specifically in the following form: Where ζ is the modal damping ratio; for multimodal coupling conditions, wavelet transform is used to extract the frequency and amplitude features corresponding to each modal ridge, and the extraction results are used as the basis for setting the parameters of the disturbance observer filter and scheduling the feedback gain; Step 5.3: The natural frequencies and mode shapes satisfy the following generalized eigenvalue equations: Where K(q) and M(q) are the pose-related stiffness matrix and mass matrix, respectively, ωi is the i-th natural frequency, and Φi is the corresponding mode shape; Step 6.1: For the dominant flexible mode that contributes significantly to the end-effector vibration response, set a target closed-loop pole to shift the flexible mode pole to the high-damping region. The specific form of the target pole is as follows: Step 6.2: Establish the constraint relationship between closed-loop poles and closed-loop eigenvectors, specifically in the form of: Step 6.3: While configuring the flexible modal poles and eigenvectors, retain the rigid modal feature structure related to rigid body trajectory tracking, so that the trajectory tracking performance of the robotic arm is not significantly weakened by the enhancement of flexible modal damping.

7. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 1, characterized in that, Step 7 describes a pose-related gain scheduling mechanism that includes the following steps: Step 7.1: Select joint variables that are sensitive to the frequency changes of the dominant flexible mode of the robotic arm as scheduling variables, and calculate the corresponding partial feature structure configuration feedback gain matrix offline under several typical joint poses; Step 7.2: Establish the interpolation scheduling relationship between the current joint pose q and the feedback gain matrix, specifically in the form of: Step 7.3: When the robotic arm pose changes, calculate the scheduling weights in real time based on the current joint pose and update K(q). s This ensures that the dominant flexible mode maintains the target closed-loop damping characteristics under different poses; Step 7.4: The pose-related gain scheduling mechanism and the inner-loop disturbance observer share the same linear nominal model and residual uncertainty constraints, thereby achieving coordinated control of disturbance compensation, flexible mode damping enhancement and pose-related mode drift compensation.

8. The method for suppressing residual vibration of an industrial robotic arm based on a disturbance observer and partial feature structure configuration according to claim 1, characterized in that, Step 8, the final fusion process of the joint drive control input, includes the following steps: Step 8.1: Obtain the trajectory tracking control quantity output by the basic servo controller to ensure that the joint position, speed and acceleration of the robotic arm track the preset trajectory; Step 8.2: Obtain the lumped disturbance estimate d output by the disturbance observer, and invert it to use as the feedforward compensation control quantity; Step 8.3: Obtain the partial feature structure configuration feedback gain after pose scheduling, and generate modal damping control quantity based on the current state variable x; Step 8.4: Fuse the basic servo control input, disturbance feedforward compensation control input, and modal damping control input to obtain the final joint drive control input, specifically in the form of: Step 8.5: Send the final joint drive control input to the robotic arm servo driver to enable the robotic arm to reduce the residual vibration amplitude at the end of the arm, shorten the settling time, reduce the overshoot, and enhance the robustness in high-speed positioning, frequent start-stop, impact response, and complex trajectory tracking tasks.

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