Mechanical arm dynamics analysis and modeling method and mechanical arm system

Abstract

The invention provides a mechanical arm dynamics analysis and modeling method and a mechanical arm system, and relates to the technical field of dynamics analys.The mechanical arm dynamics analysis and modeling method includes the steps that linear processing is conducted on a coupling dynamics model at the reference time point of a high-speed movement track, and a feedforward observer is designed; additional compensation torque used for suppressing flexible residual vibration is calculated through off-line open-loop simulation, structural vibration excited by high-speed motion is predicted and actively compensated in advance without depending on real-time feedback, the limitation that vibration cannot be suppressed in time due to hysteresis in traditional feedback control is effectively overcome, and through feed-forward vibration suppression, the control precision is improved. And the residual vibration amplitude and the stabilization time after the movement is stopped are obviously reduced, so that the high-speed performance potential of the light-weight mechanical arm is fully exerted while the tail end positioning precision is guaranteed. According to the method, the real-time performance and stability of control are improved, the online calculation burden is reduced, and integration implementation in a robot control system is facilitated.

Description

Technical Field

[0001] This invention relates to the field of dynamic analysis technology, specifically to a method for dynamic analysis and modeling of a robotic arm and a robotic arm system. Background Technology

[0002] In modern industrial fields such as electronics manufacturing, precision assembly, and high-speed sorting, SCARA (Selective Compliant Assembly Robotic Arm) robots are widely used due to their high-speed and high-precision motion characteristics in the horizontal plane. To achieve higher production cycles, the requirements for the motion speed and acceleration of the robotic arm's end effector are becoming increasingly stringent. To achieve this goal, lightweight design has become an inevitable trend in robot structural development, the core of which lies in using lightweight materials and optimizing component cross-sections to reduce moment of inertia. However, while this lightweight design brings the potential for improved dynamic performance, it also significantly reduces the structural stiffness of the robotic arm (especially the second robotic arm), causing significant elastic deformation and continuous structural vibration during high-speed start-stop and trajectory tracking due to its own flexibility. This residual vibration not only seriously affects the end-effector positioning accuracy and repeatability, reducing work quality, but also leads to prolonged system stabilization time, fundamentally restricting the full realization of high-speed performance.

[0003] Currently, the mainstream methods for dynamic control of industrial robots are still largely based on multi-rigid-body dynamics models. These models assume that all links are absolutely rigid bodies and only consider concentrated flexibility at joints (such as the stiffness of harmonic reducers). When faced with the distributed flexibility introduced by the lightweight arm itself, these models have inherent limitations. They cannot accurately describe and predict the complex vibration modes generated by flexible deformation during high-speed movement of the robotic arm. This results in controllers designed based on this model (such as feedforward control and computational torque control) being unable to generate precise torque commands that can actively counteract such vibrations. Consequently, the performance boundary of the control system is limited by the vibration characteristics of the mechanical structure. To suppress vibration, it is often necessary to actively reduce the movement speed, creating a contradictory situation where stability is sacrificed for efficiency.

[0004] The information disclosed in the background section is only intended to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide a method for analyzing and modeling the dynamics of a robotic arm and a robotic arm system, so as to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution: A method for analyzing and modeling the dynamics of a robotic arm, comprising the following steps: Step 1: For the first and second robotic arms connected by a rotary joint, construct their basic multi-rigid-body dynamics model; and for the second robotic arm, use the assumed modal method to describe its flexible deformation as a linear superposition of low-order mode shapes to construct a coupled dynamics model. Step 2: Obtain the key dynamic parameters of the second robotic arm through excitation experiments, including at least the natural frequency, mode shape function, and modal damping ratio, and input them into the coupled dynamic model; Step 3: Select a reference time point on the predetermined high-speed motion trajectory. Using the desired state at the reference time point as the benchmark, define the state variables as the deviations of each state quantity from the desired state. Linearize the coupled dynamics model at the reference time point and derive the linearized state space equation. Step 4: Design a feedforward observer based on linearized state-space equations; In the offline simulation stage, input the trajectory command of the predetermined high-speed motion trajectory into the feedforward observer, and calculate the additional compensation torque for suppressing the residual vibration of the second robotic arm through an open-loop strategy. Step 5: Perform a test on the robotic arm that includes high-speed start and stop, and collect the actual joint torque and the actual vibration acceleration at the end of the second robotic arm; at the same time, input the trajectory command of the test into the coupled dynamics model for simulation and obtain the corresponding predicted value; compare the collected actual value with the corresponding predicted value to determine whether the modal damping ratio parameter in the coupled dynamics model needs to be optimized.

[0007] Furthermore, based on the recursive Newton-Euler algorithm, the multi-rigid-body dynamics equations of the rigid running parts of the first and second robotic arms are established to construct a basic model of multi-rigid-body dynamics. The logic for describing the flexible deformation of the second robotic arm is as follows: the second robotic arm is modeled as a cantilever beam with one end fixed and the other end free; using the hypothetical modal method, the lateral bending elastic deformation of the second robotic arm in the motion plane is described as a linear superposition of low-order mode shapes, specifically including: expressing the lateral elastic displacement function of the second robotic arm along its length direction as the sum of the products of each order mode shape function and the corresponding order mode generalized coordinates; wherein, the low-order mode shapes are characterized by the bending frequency and the bending mode shape function; the mode generalized coordinates are independent generalized coordinates describing the change of the vibration amplitude of the corresponding order mode with time; the mode shape function at least includes functions corresponding to the first and second order bending modes of the second robotic arm.

[0008] Furthermore, using the second type of Lagrange equations, the multi-rigid-body dynamics equations are energetically coupled with the flexible deformation description to construct a coupled dynamics model, specifically including: Define a complete generalized coordinate vector for the robotic arm, which includes the first joint angle describing the rotation of the first joint, the second joint angle describing the rotation of the second joint, and all modal generalized coordinates describing the second robotic arm; the first joint connects the base and the first robotic arm and drives the first robotic arm to rotate, and the second joint connects the first robotic arm and the second robotic arm and drives the second robotic arm to rotate. The total kinetic energy of the robotic arm is calculated, including the kinetic energy of the first robotic arm, the kinetic energy of the second robotic arm due to the rigid motion of its center of mass, and the additional kinetic energy of the second robotic arm due to its elastic deformation. When calculating the additional kinetic energy, the second robotic arm is integrated along its length, and the kinetic energy component generated by the lateral elastic deformation velocity of each cross section is accumulated. Calculate the total potential energy of the robotic arm, including the gravitational potential energy of the first and second robotic arms, and the elastic potential energy generated by the bending deformation of the second robotic arm; Substituting the total kinetic energy and total potential energy into the second type of Lagrange equations, we obtain a second type of differential equation with the complete generalized coordinate vector as the variable. This second type of differential equation is the coupled dynamic equation; the coupled dynamic model is expressed by this coupled dynamic equation.

[0009] Furthermore, obtaining the key dynamic parameters specifically includes: Multiple acceleration sensors are arranged on the body of the second robotic arm, and an excitation is applied to its free end region. The excitation is either a pulse excitation or a broadband random excitation. The vibration response signals of each accelerometer under excitation are collected, and the excitation signal is collected simultaneously. The collected excitation and vibration response signals are processed, and the natural frequencies, mode shape functions, and modal damping ratios of the first two dominant bending modes of the second robotic arm in its motion plane are identified and extracted through frequency response function analysis or operational modal analysis. The extracted modal vibration function is directly assigned to the preset modal vibration function used when constructing the coupled dynamics model; the extracted modal damping ratio is input as a known parameter into the coupled dynamics equation to characterize the internal resistance characteristics of the material and structure of the second robotic arm.

[0010] Furthermore, for the pre-planned high-speed trajectory including high-speed motion segments, the linearization logic for the coupled dynamics model at the reference time point is as follows: The high-speed motion trajectory is a time series that describes the expected state of each joint of the robotic arm at each time point, including the expected rotation angle, expected angular velocity, and expected angular acceleration. The time point when the second joint angular velocity is the largest or the time point when the second joint angular acceleration reaches its peak value in the high-speed motion trajectory is selected as the reference time point for characterizing the high-speed dynamic characteristics. Obtain the desired rotation angle and desired angular velocity of the first and second joints at the reference time point, and set all modal generalized coordinates and their velocities of the second robotic arm to zero to form the reference state vector corresponding to the reference time point; Substituting the reference state vector into the coupled dynamic equations and setting all generalized accelerations to zero, the joint motor torque required to maintain static equilibrium under this reference state is obtained by solving the equations. This torque is called the static equilibrium torque. The state variables are the actual rotation angle and actual angular velocity of each joint, as well as the actual modal generalized coordinates and actual speed of the second robotic arm. The state variable vector is defined as the deviation of each state variable from the corresponding component in the reference state vector. The input variable vector is defined, whose elements are the deviation of the actual output torque of each joint motor from the corresponding component in the static balance torque. At the reference time point, using the above deviation as a new variable, a first-order Taylor expansion of the coupled dynamics equation is performed to derive the state matrix and input matrix describing the dynamic evolution of the deviation. Based on the state matrix and the input matrix, a linearized state-space equation is constructed at the reference time point.

[0011] Furthermore, the design of the feedforward observer specifically includes: Linearized state-space equations are used as forward prediction models to describe the rigid-flexible coupling dynamics of the robotic arm. The model is run in a feedforward open-loop mode, with the trajectory command of the high-speed motion trajectory as the expected input. The trajectory command includes at least the expected angle, expected angular velocity and expected angular acceleration of each joint at each moment, which constitutes the feedforward observer.

[0012] Furthermore, the feedforward observer recursively calculates the corresponding system state prediction sequence based on the linearized state-space equation, and the system state prediction sequence includes the predicted state of the flexible deformation of the second robotic arm. The total feedforward torque required to generate the system's predicted state sequence is calculated from the operation of the feedforward observer; the part directly related to suppressing flexible deformation is separated from this total feedforward torque and defined as the additional compensation torque. The additional compensation torque is a time-varying sequence whose amplitude and frequency characteristics are related to the acceleration variation of the high-speed motion trajectory and the natural frequency of the second robotic arm. This additional compensation torque is ultimately superimposed on the torque command of the joint motor to actively counteract the structural vibrations excited by the high-speed motion.

[0013] Furthermore, the test experiment including high-speed start and stop shall at least meet one of the following conditions: the maximum linear velocity of the end effector of the second robotic arm during the movement process is not less than 2 m / s, and the absolute value of the joint angular acceleration within the movement segment of the test trajectory obtained through the test experiment is not less than 100 radians per square second. When controlling the robotic arm to actually execute the test trajectory, the actual joint torque converted from the actual output current of the joint motors driving the first and second robotic arms is collected simultaneously, as well as the actual vibration acceleration measured by the accelerometer installed on the end effector of the second robotic arm.

[0014] Furthermore, determining whether to optimize the modal damping ratio specifically includes: The trajectory commands corresponding to the test including high-speed start and stop are input into the coupled dynamics model for numerical simulation, and the predicted total joint torque sequence and predicted vibration acceleration sequence of the simulation output are obtained. The torque error between the actual joint torque and the predicted total joint torque sequence is compared, and the difference in decay characteristics between the actual vibration acceleration and the predicted vibration acceleration sequence after the motion stops is analyzed; the decay characteristics are characterized by the decay rate. If the torque error or the difference in attenuation characteristics exceeds the preset threshold, the modal damping ratio in the coupled dynamics model will be increased or decreased accordingly, depending on whether the attenuation rate of the actual vibration acceleration is faster or slower than the attenuation rate of the predicted vibration acceleration. The updated modal damping ratio is re-introduced into the coupled dynamics model, and the above steps are iteratively executed until the torque error and attenuation characteristic difference do not exceed the preset threshold. The last updated modal damping ratio parameter is then used in the subsequent control of the robotic arm.

[0015] The present invention also provides a robotic arm dynamics analysis and modeling system, which is used to implement the above-mentioned robotic arm dynamics analysis and modeling method, including: A coupled dynamics model module is established to construct a multi-rigid-body dynamics basic model for the first and second robotic arms connected by a rotary joint; and for the second robotic arm, the assumed modal method is used to describe its flexible deformation as a linear superposition of low-order mode shapes to construct a coupled dynamics model. The experimental modal parameter acquisition module is used to obtain the key dynamic parameters of the second robotic arm through excitation experiments, including at least the natural frequency, mode shape function and modal damping ratio, and input them into the coupled dynamic model; The state space equation generation module is used to select a reference time point on a predetermined high-speed motion trajectory, use the desired state at the reference time point as a benchmark, define the state variables as the deviations of each state quantity relative to the desired state, linearize the coupled dynamics model at the reference time point, and derive the linearized state space equation. The compensation torque calculation module is used to design a feedforward observer based on the linearized state-space equation. During the offline simulation stage, the trajectory command of the predetermined high-speed motion trajectory is input into the feedforward observer, and the additional compensation torque used to suppress the residual vibration of the second robotic arm is calculated through the open-loop method. The damping parameter iterative calibration module is used to perform tests including high-speed start and stop on the robotic arm, collect the actual joint torque and the actual vibration acceleration at the end of the second robotic arm; at the same time, the trajectory command of the test is input into the coupled dynamics model for simulation to obtain the corresponding predicted value; the collected actual value is compared with the corresponding predicted value to determine whether the modal damping ratio parameter in the coupled dynamics model needs to be optimized.

[0016] The technical effects and advantages provided by the present invention in the above technical solution are as follows: This invention achieves accurate prediction and active suppression of the flexible residual vibration of a robotic arm under high-speed motion, thereby fully releasing the high-speed dynamic performance potential inherent in the lightweight structure while ensuring the positioning accuracy and stability of the end effector. Specifically, the established rigid-flexible coupled dynamic model, compared with the traditional multi-rigid-body model, can more realistically reflect the distributed flexibility characteristics of the second robotic arm, laying a precise model foundation for vibration analysis and suppression. Based on excitation experiments, the modal parameters of the actual structure (especially the damping ratio, which is difficult to determine theoretically) are obtained and injected, significantly improving the model's fidelity and prediction accuracy. Furthermore, by linearizing the model for control guidance and designing a feedforward observer, the compensation torque required to counteract the vibration excited by a specific high-speed trajectory can be calculated in advance, realizing feedforward active suppression of vibration and solving the problem of hysteresis in traditional feedback control.

[0017] In summary, this invention effectively solves the core contradiction of being unable to achieve both high speed and high precision due to inaccurate models, control lag, and lack of optimization. It enables the robotic arm to maintain high-precision operation while operating stably in a higher speed range, significantly improving production cycle time and overall system performance. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the overall method flow of the present invention; Figure 2 This is a schematic diagram comparing the optimized joint torque data of the present invention; Figure 3 This is a schematic diagram comparing the optimized vibration acceleration data of the present invention; Figure 4 This is a schematic diagram of the robotic arm system structure of the present invention. Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to specific embodiments.

[0020] It should be noted that, unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0021] Example: Please see Figures 1 to 3 The present invention provides a technical solution: A method for analyzing and modeling the dynamics of a robotic arm, comprising the following steps: Step 1: For the first and second robotic arms connected by a rotary joint, construct their basic multi-rigid-body dynamics model; and for the second robotic arm, use the assumed modal method to describe its flexible deformation as a linear superposition of low-order mode shapes to construct a coupled dynamics model.

[0022] For the first robotic arm (rigid arm) and the second robotic arm (flexible arm) connected by a rotary joint, a basic multi-rigid-body dynamics model is first constructed. This model treats the rigid running parts of the two robotic arms as ideal rigid bodies, ignores the elastic deformation of the second robotic arm, and only describes the rigid body motion driven by the joint motor.

[0023] In this embodiment, the recursive Newton-Euler algorithm is used to establish the model. This algorithm is an efficient, recursive calculation method that calculates the velocity and acceleration of each link outward from the base, and then calculates the joint forces and torques inward from the ends, ultimately deriving the dynamic equations of the system. The recursive nature of this algorithm makes it particularly suitable for tree-like or chain-like multibody systems. The computational complexity is linearly related to the number of joints, resulting in high efficiency.

[0024] Besides the recursive Newton-Euler method, those skilled in the art can also use the Lagrange method to establish this basic model. For example, first define the generalized coordinates of the system (such as the rotation angles of the two joints), calculate the total kinetic energy and total potential energy of the system respectively, and then substitute them into the second kind of Lagrange equations to derive the multi-rigid-body dynamics equations. The Lagrange method is based on the principle of energy, has clear physical meaning, and is particularly convenient for introducing additional energy terms such as flexible deformation into the system.

[0025] For accurate modeling, the flexibility of the second robotic arm must be taken into account. In this embodiment, the second robotic arm is modeled as a cantilever beam with one end fixed to the joint connecting it to the first robotic arm (i.e., the second joint) and the other end free. Its flexible deformation is mainly considered as lateral bending in the plane of motion.

[0026] The assumed modal method is used to describe its elastic deformation. This method approximates the deformation of a continuous, infinite-degree-of-freedom elastic beam as a linear superposition of a finite number of known, independent mode shapes. This is a model order reduction technique that can capture the dominant flexible dynamic characteristics with low computational cost. Specifically, the lateral elastic displacement function of the second robotic arm along its length is expressed as the sum of the products of the mode shape functions of each order and the corresponding generalized coordinates of the modes.

[0027] To balance computational accuracy and efficiency, this embodiment includes functions corresponding to at least the first and second order bending modes of the second robotic arm, meaning the selected modal order is at least 2. The specific value of the modal order is typically determined through pre-experiments or finite element modal analysis to ensure that the selected modes cover the main dynamic responses within the operating frequency range. For example, if the main component of the excitation frequency of the second robotic arm under the planned high-speed motion trajectory is lower than the third natural frequency, then the modal order can be set to 2; if there is a risk of higher frequency excitation, the modal order should be increased.

[0028] The modal shape function is a predefined function about position, describing the deformation shape under each mode. For a cantilever beam with a uniform cross-section, its analytical solution is a known Euler-Bernoulli beam mode shape function. For cantilever arms with non-uniform or complex structures, the modal shape function can be obtained at discrete points through finite element analysis or experimental modal analysis in subsequent step 2, and then constructed into a continuous function through interpolation (such as cubic spline interpolation).

[0029] Modal generalized coordinates are independent generalized coordinates that vary with time and describe the magnitude of the vibration amplitude of the corresponding order mode. These modal generalized coordinates will become new state variables added to the coupled dynamics model.

[0030] The low-order mode shape is characterized by the corresponding bending frequency and bending mode shape function. The bending frequency determines the speed of vibration of the mode, and the mode shape function determines the spatial distribution of the vibration.

[0031] In this embodiment, the Lagrange second kind equation is used to couple the multi-rigid-body dynamics equations with the flexible deformation description to construct a coupled dynamics model, specifically including: Define a complete generalized coordinate vector for the robotic arm, which includes the first joint angle describing the rotation of the first joint, the second joint angle describing the rotation of the second joint, and all modal generalized coordinates describing the second robotic arm. The first joint connects the base and the first robotic arm, driving the first robotic arm to rotate; the second joint connects the first robotic arm and the second robotic arm, driving the second robotic arm to rotate.

[0032] The total kinetic energy of the robotic arm consists of three parts: The kinetic energy of the first robotic arm: Since it is a rigid body, its kinetic energy can be directly calculated from its translational velocity of the center of mass and its rotational velocity around the center of mass.

[0033] The kinetic energy generated by the rigid motion of the second robotic arm's center of mass: Treating the second robotic arm as a rigid body, calculate the translational and rotational kinetic energy generated by the joint motion of its center of mass.

[0034] The additional kinetic energy generated by the elastic deformation of the second robotic arm is crucial for the coupling. The calculation requires integrating along the length of the second robotic arm and summing the kinetic energy components generated by the lateral elastic deformation velocity at each of its infinitesimal cross-sections. (The calculation of the additional kinetic energy involves integrating along the length of the second robotic arm and summing the kinetic energy components generated by the lateral elastic deformation velocity at each of its cross-sections.)

[0035] The total potential energy of the robotic arm includes: The gravitational potential energy of the first and second robotic arms is determined by the height of their centers of mass. The elastic potential energy generated by the bending deformation of the second robotic arm: This is generated by the bending deformation of the second robotic arm. According to mechanics of materials, the elastic potential energy of an Euler-Bernoulli beam can be calculated using the strain energy formula based on beam bending theory.

[0036] Substituting the total kinetic energy and total potential energy into the standard form of the Lagrange equation of the second kind, we obtain a second kind of differential equation with the complete generalized coordinate vector as the variable. This second kind of differential equation is the coupled dynamics equation. This equation describes the dynamic characteristics of the system and, through simplification, can be expressed as a matrix containing the inertia matrix, Coriolis force and centrifugal force matrices, stiffness matrix, gravity vector, and input matrix. For modal coordinates, if there is no external lateral force, the generalized force is zero. Through the above derivation, we finally obtain a system of second-order differential equations with the complete generalized coordinate vector as the variable, i.e., the coupled dynamics equation. The stiffness matrix, mainly derived from the elastic potential energy, is a diagonal or block diagonal matrix, whose diagonal elements are proportional to the square of the natural frequencies of each mode.

[0037] This equation clearly reveals the complete dynamic coupling process of how joint motion excites flexible modes and how the vibration of flexible modes reacts to joint motion, which is the basis for all subsequent analysis and control design.

[0038] Before conducting the excitation experiment in step 2, if the acquired raw vibration response signal contains significant noise or DC offset, preprocessing is required. Preprocessing typically includes: Detrending: Removing slowly changing trend terms from the signal (e.g., caused by temperature drift or sensor zero-point drift). Common methods include subtracting the low-order polynomial fit or moving average of the signal. Bandpass filtering: Designing and applying a digital bandpass filter (such as a Butterworth filter) based on the desired modal frequency range (e.g., 0.5 times the lowest target modal frequency to 1.5 times the highest target modal frequency) to retain the effective frequency band signal and suppress high-frequency noise and low-frequency interference. Outlier removal: Identifying and eliminating abnormal amplitude points (outliers) caused by measurement interference, which can be achieved using median filtering or methods based on statistical thresholds.

[0039] Step 2: Obtain the key dynamic parameters of the second robotic arm through excitation experiments, including at least the natural frequency, mode shape function and modal damping ratio, and input them into the coupled dynamic model.

[0040] Natural frequency: refers to the specific frequency at which the second robotic arm vibrates freely under its constrained conditions (one end fixed, the other end free). The first two natural frequencies (usually the lowest two) determine the frequency points at which the flexible arm is most likely to resonate when subjected to external excitation. They are important bases for designing controllers (such as the feedforward observer in step 4) to avoid exciting resonance and for designing filters.

[0041] Modal mode functions describe the relative shape and amplitude ratio of the vibration displacement at various points along the length of a flexible arm at a specific natural frequency. When constructing the coupled dynamics model, we predetermine the modal mode functions (such as orthogonal function sets derived from Euler-Bernoulli beam theory). However, the theoretically predetermined mode functions deviate from the physical entity due to manufacturing tolerances, material inhomogeneities, and added mass (such as cables and sensors). Assigning experimentally identified true mode functions to the model is crucial to ensuring that the model closely resembles the physical system.

[0042] Modal damping ratio: A dimensionless parameter characterizing the rate at which energy dissipates during vibration of a flexible arm structure (caused by internal material friction, joint friction, air resistance, etc.). It directly affects the vibration decay rate and amplitude predicted by the model. Theoretical models typically cannot accurately preset this parameter and must be determined experimentally.

[0043] In this embodiment, obtaining the key dynamic parameters specifically includes: Multiple acceleration sensors are arranged on the body of the second robotic arm. The arrangement principle should ensure that the mode shape characteristics of the first two bending modes to be identified can be effectively captured.

[0044] For a second robotic arm of length L, a single-axis accelerometer can be installed at distances of 0.2L, 0.4L, 0.6L, 0.8L, and 1.0L (free end) from the fixed end (joint), along the normal direction of the motion plane. This arrangement, whether equally spaced or not (e.g., denser near mode shape nodes), provides sufficient spatial sampling points for subsequent fitting of continuous mode shape functions. Lightweight piezoelectric or MEMS accelerometers with a frequency response range covering the first two natural frequencies (estimated values, e.g., 0-200Hz) should be selected to minimize the impact of added mass on the dynamic characteristics of the structure under test.

[0045] An excitation is applied to the free end region of the second robotic arm, and the excitation should have sufficient bandwidth to cover the first two natural frequencies.

[0046] For pulse excitation, an impact hammer equipped with a force sensor strikes a designated point near the free end. The hammerhead material (e.g., rubber, plastic, steel) can be adjusted to control the spectral width of the impact force. This method is simple and fast, but requires multiple strikes to ensure data repeatability. For broadband random excitation, a small exciter is connected to the free end via a low-stiffness spring or flexible rod, and a signal generator produces white noise or pseudo-random signals to drive the exciter. This method has the advantages of uniform excitation energy distribution across the frequency band, high signal-to-noise ratio, and more robust identification results. Vibration response signals from all accelerometers and excitation signals (force signal from the impact hammer or drive signal from the exciter) must be acquired simultaneously. The sampling frequency of the data acquisition equipment should be at least five times the highest natural frequency to be measured to satisfy Shannon's sampling theorem and ensure identification accuracy.

[0047] The vibration response signals of each accelerometer under excitation are collected, and the excitation signal is collected simultaneously.

[0048] The acquired excitation and vibration response signals are processed to extract key parameters, as detailed below: For each accelerometer measurement point, the frequency response function between its vibration acceleration signal and excitation force signal is calculated. The frequency response function is usually calculated using the average periodogram method based on the fast Fourier transform (such as the H1 or H2 estimator) to reduce the influence of noise on the analysis results.

[0049] Extraction of natural frequencies: By observing the peak positions of the amplitude spectrum of the frequency response function of all measuring points, if each measuring point has a significant peak at a certain frequency, then that frequency is identified as the natural frequency of a certain mode (for example, for the first two dominant bending modes, they are the first natural frequency and the second natural frequency, respectively).

[0050] Modal damping ratio extraction: For the identified nth mode, the modal damping ratio can be extracted from the amplitude spectrum of the frequency response function using the half-power bandwidth method. Specifically, at the peak value corresponding to the natural frequency of this mode, find two frequency points where the amplitude drops to approximately 0.707 times the peak value (i.e., a decrease of approximately 3 dB). These are denoted as the lower bound frequency and the upper bound frequency, respectively. The modal damping ratio can then be approximately calculated by dividing the difference between these two frequency points by twice the natural frequency.

[0051] Modal mode function extraction and fitting: At the natural frequency, the relative proportions of the phase (usually close to 0° or 180°) and amplitude of the frequency response function at each measuring point together constitute the discrete mode vector of that order at the sensor placement location. Subsequently, this discrete vector needs to be fitted into a preset mode function form that is continuous along the length of the robotic arm. This fitting process is usually based on a theoretically preset mode function basis (such as polynomial basis functions or standard cantilever beam mode function). The least squares method is used to fit the experimental discrete mode data to the preset function basis, thereby obtaining an experimentally corrected, continuous modal mode function.

[0052] When it is impossible to accurately measure the input excitation (such as environmental excitation or when force sensors cannot be installed) by only measuring the response (e.g., using only an accelerometer), operational modal analysis can be used. This involves collecting response data from each sensor under excitation, and estimating the system matrix, poles, and residues directly from the time series or cross-power spectrum of the response data using methods such as random subspace identification or frequency domain decomposition. This allows the natural frequencies, damping ratios, and mode shapes to be obtained.

[0053] The modal shape functions identified and fitted through experiments are directly and completely replaced with the preset mode shape functions used in constructing the coupled dynamics model. This step is crucial for improving the accuracy of the model's representation of the actual structural dynamics. Simultaneously, the identified modal damping ratios are explicitly introduced into the coupled dynamics equations obtained in step 1 as known parameters, in the form of damping matrices or Rayleigh damping coefficients. For example, in the general form of the second-order differential equation derived from the Lagrange equation, the elements on the diagonal of the damping matrix corresponding to each modal coordinate can be constructed according to the product relationship of the modal damping ratio, the natural circular frequency of that mode, and the modal mass. The accurate introduction of the modal damping ratio enables the model to more realistically predict vibration attenuation characteristics consistent with the actual structure. In this equation, the first two elements of the generalized force vector correspond to the torques applied by the first and second joint motors, while the remaining elements related to the flexible modes are usually set to zero, indicating that the excitation of the flexible modes is indirectly achieved through the dynamic coupling of joint motion.

[0054] Step 3: Select a reference time point on the predetermined high-speed motion trajectory. Using the desired state at the reference time point as the benchmark, define the state variables as the deviations of each state quantity from the desired state. Linearize the coupled dynamics model at the reference time point and derive the linearized state-space equation.

[0055] In this embodiment, for a pre-planned high-speed trajectory including a high-speed motion segment, the logic for linearizing the coupled dynamics model at the reference time point is as follows: The high-speed motion trajectory is a pre-planned spatiotemporal path in the controller that drives the robotic arm to complete a specific task (such as rapid pick-and-place, high-speed spraying, etc.); it includes the expected rotation angle, expected angular velocity, and expected angular acceleration of each joint at each moment. It is also a complete time series, for example, stored discretely in the controller memory with a fixed control period (such as 1ms).

[0056] Linearization is performed on the trajectory points where the flexible vibration is most severely excited and the system nonlinearity is most significant, to ensure that the linear model can still maintain high approximation accuracy under this worst-case condition.

[0057] In principle, the time point with the maximum angular velocity of the second joint or the time point where the angular acceleration of the second joint reaches its peak during high-speed motion can be selected as a reference time point to characterize high-speed dynamic characteristics. The point of maximum angular velocity of the second joint is chosen because the Coriolis force and centrifugal force generated by high-speed motion are the main excitation sources for the residual vibration of the flexible arm. The greater the angular velocity of the second joint (connecting the flexible arm), the greater these inertial forces, and the more significant their coupling effect on flexible deformation. Linearization at this point best captures the dynamic characteristics of velocity-related terms. The peak point of the angular acceleration of the second joint is chosen because joint acceleration directly corresponds to the driving torque, which is the direct input of vibration excitation. The peak acceleration point signifies the most drastic change in inertial load and the moment when energy injection into the flexible mode is most intense. Linearization at this point most accurately describes the dynamic relationship between input (torque) and output (state, especially the flexible mode).

[0058] For example, after planning the trajectory, the system will traverse the trajectory data sequence to find the time point that makes the absolute value of the expected angular velocity of the second joint reach the global maximum. And the time point at which the absolute value of the desired angular acceleration of the second joint reaches its global maximum. One point can be selected as the sole reference time. A more preferred implementation involves performing linearization twice, using these two moments as reference points, and designing two corresponding feedforward observers. During actual control, these observers are switched or merged based on the current motion state (approaching a high-speed segment or a high-acceleration segment). For simplicity, a single reference point will be used below. Let's take an example to illustrate this.

[0059] The reference state vector is a complete and specific numerical description of the ideal desired state of all moving parts of the robotic arm at that reference time point; it constitutes the zero point or scale for all our subsequent deviation calculations.

[0060] Specifically, establishing this vector involves the following explicit steps: First, extract the command data for the selected specific reference time point (e.g., the millisecond when the second joint angular acceleration reaches its maximum) from the pre-planned high-speed motion trajectory commands. This data explicitly gives the ideal angle values ​​that the first and second joints should achieve at this time, as well as the ideal rotational speed values ​​they should possess. These two angles and two speeds are the first four core components of the reference state vector.

[0061] Second, regarding the elastic deformation state of the second robotic arm (flexible arm), when constructing this idealized reference state, we make a crucial and reasonable physical setting: assuming that at this moment, the flexible arm has no bending vibration. This means that the independent coordinate value used to describe its first-order bending deformation amplitude and its rate of change should both be set to zero; similarly, the coordinate value used to describe its second-order bending deformation amplitude and its rate of change should also be set to zero. If higher-order bending modes are considered in the modeling, their corresponding amplitude and velocity coordinates are also set to zero.

[0062] Arranging the aforementioned components—the ideal angles of the two joints, the ideal velocities of the two joints, and the zero amplitude and zero velocity of all flexible modes—in sequence creates a complete number vector, which forms the reference state vector. This vector precisely depicts the ideal scenario at that high-speed instant when the robotic arm moves entirely according to instructions without any elastic vibration.

[0063] The logical process for calculating this static equilibrium torque is as follows: First, the specific reference state vector obtained above, which includes the specific joint angles, joint velocities, and all zero flexible deformation coordinates, is substituted into the complete coupled dynamics model (i.e., the mathematical equation set that simultaneously considers the rotation of the rigid body and the bending of the flexible beam) that we established in step one. Second, a core freezing assumption is applied: since we are calculating the force required to maintain this state, we assume that the state of the robotic arm remains exactly at this moment and does not evolve to the next moment. Therefore, in this calculation scenario, the acceleration of all variables is considered zero. This includes the joint angular acceleration being zero and the vibration acceleration of the flexible modes being zero.

[0064] Next, the mathematical solution is performed. When the specific state values ​​(angle, velocity) and zero acceleration are substituted into the dynamic equations, the terms describing the inertial forces of mass (because the acceleration is zero) and the terms describing the elastic restoring forces (because the deformation amplitude is set to zero) will disappear or be simplified. At this point, the equations mainly consist of terms describing the inertial effects (such as centrifugal force and Coriolis force) caused by high-speed rotation, as well as terms that overcome the effects of gravity.

[0065] Finally, by solving this simplified system of algebraic equations, the torque values ​​required for the two articulated motors to precisely balance the aforementioned inertial and gravitational forces under this specific high-speed motion state can be directly determined. These two calculated torque values ​​constitute the static equilibrium torque. This torque is a specific, calculable numerical result, serving as the reference force for subsequent control command design. The actual control commands will be based on this reference force, with the addition of a corrective force to adjust for minor state deviations.

[0066] The state variables are the actual rotation angle and actual angular velocity of each joint, as well as the actual modal generalized coordinates and actual speed of the second robotic arm. Here, each actual state variable refers to the one obtained from simulation calculations performed by the coupled dynamics model under given inputs. The state variable vector is defined as the deviation of each state variable from the corresponding component in the reference state vector. For example, the angle deviation of the first joint is obtained by subtracting the expected angle of its reference point from the actual angle of the first joint; the deviation of the first-order modal coordinates is obtained by subtracting zero from the actual first-order modal coordinates of the second robotic arm. Similarly, a deviation is defined for each state variable (angle, velocity, modal coordinates, and speed). The input variable vector is defined, whose elements are the deviations between the actual output torque of each joint motor and the corresponding component in the static balance torque.

[0067] In mathematics, these deviation variables are used as new objects of analysis. At the precise operating point determined by the reference state and static equilibrium torque, a mathematical approximation called a first-order Taylor expansion is performed on the original, complex nonlinear dynamic equations. The core idea of ​​this process is that when the deviation between the actual operating state of the system and the reference point is extremely small, the complex nonlinear behavior of the system can be approximated as a combination of these deviation quantities in a simple proportional (i.e., linear) relationship. After expansion, all higher-order multiplication terms of the deviation quantities (such as the square of the deviation, the product of two deviations, etc.) are ignored because their values ​​are extremely small. Based on the state matrix and the input matrix, a linearized state-space equation at the reference time point is constructed.

[0068] After the above expansion and organization, a highly regular linear mathematical model is finally obtained. This model explicitly states that the rate of change of all state deviations is equal to the state matrix multiplied by the current state deviation vector, plus the input matrix multiplied by the current torque deviation vector.

[0069] The state matrix is ​​a numerical table that is entirely determined by the physical properties of the robotic arm at the reference time point, such as its specific configuration, mass distribution, stiffness, and damping. It encapsulates the inherent dynamic characteristics of the robotic arm at that high-speed instant, such as the natural frequency of the flexible arm and the inertial coupling relationship between joints.

[0070] The input matrix is ​​also a numerical table that describes how effectively the torque deviation of each joint can affect the changes in various state deviations, indicating the direct path of the control action.

[0071] The effectiveness of the linearized model described above is limited; it is accurate only within a region where the deviation between the actual system state and the reference state is sufficiently small. To quantify this "sufficiently small," the concept of a deviation effectiveness threshold needs to be introduced.

[0072] The validity threshold for this deviation is not a universal fixed value, but should be set according to the specific requirements of the robotic arm and control precision. The preferred determination logic is as follows: In the computer simulation environment, the previously established nonlinear model is used as a substitute for the real system. Near the selected reference time point, a series of typical disturbances of different magnitudes are artificially applied to the system (e.g., a small initial position offset to the joint, or the application of a brief pulse torque), and simulations are performed using both the nonlinear model and the linearized model obtained in this step. The key performance indicators output by the two models are compared, such as the vibration amplitude of the second robotic arm end effector.

[0073] For example, in a second robotic arm example with an arm length of 0.5 meters, simulation comparison revealed that when the absolute value of the first joint angle deviation is less than 0.08 radians and the absolute value of the second joint angle deviation is less than 0.05 radians, the error in the end-effector residual vibration amplitude predicted by the linearized model and the nonlinear model is less than 15%. Therefore, in this embodiment, this can be used as a preliminary threshold for determining the validity of the deviation, i.e., the deviation validity threshold.

[0074] When the amplitude of the disturbance increases to a certain critical point, the prediction results of the linear and nonlinear models begin to show a non-negligible difference (for example, the prediction error of the end-effector vibration amplitude exceeds 20% of the system's allowable positioning accuracy). At this point, the magnitude of the disturbance amplitude can be used as a reference for the deviation effectiveness threshold. For example, simulations have shown that when the first joint angle deviation exceeds 0.1 radians, the predicted end-effector residual vibrations of the two models differ significantly. Therefore, in actual control, 0.1 radians can be used as an important monitoring threshold.

[0075] This method of determining thresholds based on model comparison simulation is a rigorous and universally applicable practice in engineering. It starts from mathematical principles (locality of Taylor expansion) and is directly linked to the performance indicators (control accuracy) of the actual system, providing an objective and operable standard for judging when a linear model is applicable and when it may fail.

[0076] Step 4: Design a feedforward observer based on the linearized state-space equation; during the offline simulation stage, input the trajectory command of the predetermined high-speed motion trajectory into the feedforward observer, and calculate the additional compensation torque for suppressing the residual vibration of the second robotic arm through an open-loop strategy.

[0077] In this embodiment, the feedforward observer is essentially an open-loop prediction model based on linearized state-space equations. It does not rely on the real-time feedback signal of the actual system, but rather performs calculations entirely based on pre-planned high-speed motion trajectory instructions. The design logic of the feedforward observer is as follows: The linearized state-space equation derived in step 3 at the reference time point is directly used as the core mathematical model of the observer. This equation describes how the system's state deviations (including joint angle deviations and vibration amplitude deviations of flexible modes) dynamically evolve with the input torque deviations near the reference operating point.

[0078] The input signal consists of a complete time-series data, which includes at least the expected angle, expected angular velocity, and expected angular acceleration of the first joint at each planned moment; and the expected angle, expected angular velocity, and expected angular acceleration of the second joint. These instructions represent the ideal motion path and dynamic characteristics that the robotic arm controller plans for the robotic arm to execute.

[0079] The model is run in a feedforward open-loop manner. Specifically, this involves recursively and progressively calculating the desired angular velocity, desired angular acceleration, and other information from the trajectory commands, according to the dynamic relationships defined by the linearized state-space equations. For example, given the predicted state deviation at the current moment and the input (torque deviation) calculated based on the trajectory commands, the predicted state deviation at the next moment can be calculated using the state-space equations. In this way, before the robotic arm actually moves, the predicted sequence of flexible deformations (i.e., vibration amplitudes of each mode) that the second robotic arm may undergo during the entire high-speed motion can be fully pre-enacted in computer simulation.

[0080] In this embodiment, the feedforward observer recursively calculates the corresponding system state prediction sequence based on the linearized state space equation. The system state prediction sequence includes the predicted state of the flexible deformation of the second robotic arm. The feedforward observer not only predicts the vibration state, but more importantly, it can generate control quantities to actively suppress vibration.

[0081] During the open-loop simulation, in order to drive the aforementioned state prediction model and ensure that its state (especially the state of the flexible mode) follows the excitation implied by the high-speed trajectory command, the observer will synchronously calculate a required total feedforward torque sequence. This total feedforward torque is the combined control torque theoretically required to achieve ideal rigid body motion (defined by the trajectory command) and to counteract the predicted effects of flexible vibration.

[0082] From the total feedforward torque mentioned above, the portion directly related to suppressing flexible deformation is separated. One specific implementation is as follows: First, assuming the second robotic arm is a completely rigid arm, the rigid feedforward torque required to track the same trajectory is calculated solely based on a multi-rigid-body dynamics model. Then, the total feedforward torque calculated by the feedforward observer is subtracted from this rigid feedforward torque; the difference is defined as the additional compensation torque. This torque is specifically designed to counteract the structural bending vibrations of the second robotic arm induced by dynamic effects such as acceleration during high-speed motion.

[0083] The additional compensation torque is a time-varying sequence whose amplitude, phase, and frequency characteristics are closely related to two key factors: Dynamic characteristics of high-speed motion trajectory: especially the angular acceleration variation law of the second joint, rapid acceleration and deceleration are the main sources of flexible vibration.

[0084] The inherent dynamic characteristics of the second robotic arm are the natural frequencies of the first two dominant bending modes identified in step 2. The oscillation frequency of the additional compensating torque will contain these natural frequency components, aiming to generate a force opposite to the vibration trend, thereby achieving active damping and cancellation of the vibration.

[0085] The specific implementation method is as follows: After calculating the additional compensation torque sequence in the offline simulation stage, in actual control, it is used as a feedforward compensation term and superimposed with the feedback control torque calculated based on position / velocity feedback and the rigid feedforward torque based on rigid body dynamics to form the final torque command sent to the joint motor. Formulated as: Final torque command = Feedback control torque + Rigid feedforward torque + Additional compensation torque.

[0086] This predictive vibration suppression step does not involve monitoring and suppressing vibrations after they occur via sensors, but rather predicting how the vibration will be generated using a mathematical model before the motion begins, and simultaneously generating additional compensating torque. This method is particularly suitable for high-speed, high-acceleration motion scenarios, significantly reducing residual vibration time after motion stops, improving positioning accuracy and cycle time, while avoiding the instability risks that high-gain feedback control may bring. The entire process is completed in offline simulation, without increasing the online computational burden, and is easy to integrate and implement on existing robot controllers.

[0087] Step 5: Perform a test on the robotic arm that includes high-speed start and stop, and collect the actual joint torque and the actual vibration acceleration at the end of the second robotic arm; at the same time, input the trajectory command of the test into the coupled dynamics model for simulation and obtain the corresponding predicted value; compare the collected actual value with the corresponding predicted value to determine whether the modal damping ratio parameter in the coupled dynamics model needs to be optimized.

[0088] In this embodiment, to fully stimulate the flexible dynamic characteristics of the second robotic arm and obtain effective verification data, the relevant tests must have sufficient dynamic excitation intensity. Specifically, the test experiment including high-speed start-stop must meet at least one of the following conditions: the maximum linear velocity of the second robotic arm's end effector during movement is not less than 2 m / s; this speed threshold is set based on the commonly used working speed range of the robotic arm's end effector in typical application scenarios such as high-speed industrial pick-and-place and spraying. Below this speed, flexible vibration may not be obvious, which is not conducive to parameter identification. The absolute value of the joint angular acceleration within the motion segment of the test trajectory obtained through the test experiment is not less than 100 radians per second squared; high acceleration is the main factor in stimulating structural vibration, and this threshold ensures that the motion command can significantly excite the flexible modes of the second robotic arm.

[0089] A typical and effective test trajectory is a point-to-point high-speed start-stop motion. For example, a robotic arm is instructed to accelerate from an initial stationary position A with maximum acceleration, travel a constant speed for a distance, decelerate with maximum deceleration, and finally stop precisely at the target position B, remaining stationary at B for at least 2 seconds. The entire motion should be as fast as possible to ensure the end effector reaches a high speed (satisfying condition A), and the acceleration / deceleration phases must necessarily produce high angular acceleration (satisfying condition B). The stationary phase is used to observe the free decay of vibration, which is crucial for evaluating the damping ratio.

[0090] When controlling the robotic arm to actually execute the test trajectory, the actual joint torque converted from the actual output current of the joint motors driving the first and second robotic arms is collected simultaneously, as well as the actual vibration acceleration measured by the accelerometer installed on the end effector of the second robotic arm.

[0091] Actual joint torque: By reading and recording the actual output current of the joint motors driving the first and second robotic arms, and converting it into actual joint torque according to the motor torque constant, this is the output feedback quantity of the control system.

[0092] Actual vibration acceleration: Measured and recorded by a triaxial accelerometer (which must at least measure the lateral acceleration in the plane of motion) mounted on the end effector of the second robotic arm. This is a physical quantity that directly reflects the flexible residual vibration.

[0093] The trajectory commands of the actual test trajectory (i.e., the desired angle, angular velocity, and angular acceleration time series) are used as inputs to the constructed coupled dynamics model for full digital numerical simulation. The simulation should be performed using numerical methods for solving ordinary differential equations, such as the fourth-order Runge-Kutta method.

[0094] Through simulation, two key predicted sequences were obtained: Predicted total joint torque sequence: The total torque theoretically required for each joint to track the test trajectory, calculated by the model (including all components for balancing gravity, generating acceleration, and overcoming vibration coupling forces).

[0095] Predicted vibration acceleration sequence: The model calculates the lateral acceleration sequence of the second robotic arm end in the plane of motion based on its built-in flexible deformation description (mode shape function, natural frequency, and initial modal damping ratio used in the current test).

[0096] In this embodiment, determining whether to optimize the modal damping ratio specifically includes: The torque errors of the actual joint torque and the predicted total joint torque sequence are compared, and the differences in the decay characteristics of the actual vibration acceleration and the predicted vibration acceleration sequence after motion cessation are analyzed. The decay characteristics are characterized by the decay rate, focusing on data from the period after the test trajectory is completed and the robotic arm enters a stationary phase (e.g., 2 seconds in the previous example). The signals of the actual vibration acceleration and the predicted vibration acceleration during this phase are extracted separately, filtered (e.g., bandpass filtering to remove frequency components unrelated to the flexible mode), and the decay rate of their amplitude envelope over time is observed. A simple quantification method is to calculate the exponential decay time constant of the envelope, or to directly compare the amplitude at a fixed moment (e.g., 0.5 seconds after stopping).

[0097] If the torque error is below its preset threshold, and the difference between the actual vibration acceleration decay characteristics (decay rate) and the prediction is also within its preset threshold, then the current coupled dynamics model (including its modal damping ratio parameter) is considered sufficiently accurate and requires no optimization; it can be directly used for subsequent control. If the torque error or the difference in decay characteristics exceeds the preset threshold, then the modal damping ratio value in the coupled dynamics model is increased or decreased accordingly based on whether the decay rate of the actual vibration acceleration is faster or slower than the decay rate of the predicted vibration acceleration.

[0098] The torque error threshold is used to determine the accuracy of the predicted torque, and this threshold is typically set to 3% to 10% of the rated torque of the joint motor. For example, if the rated torque of the joint motor is 100... The threshold can then be set to 3. Up to 10 This value is chosen because the percentage of rated torque is a normalized index related to system size. If the error is less than this range, it can be considered that the model's torque prediction accuracy meets the requirements of most feedforward control systems; if the error is too large, it indicates that the model's dynamic coupling terms are not accurately predicted, which may be due to inaccurate damping parameters.

[0099] The damping characteristic difference threshold is used to determine the accuracy of the predicted vibration damping rate. This threshold is typically set by comparing the amplitude ratio of the actual and predicted vibration envelopes at a specific time point after the motion stops; for example, one can examine the difference between the actual vibration acceleration amplitude at 0.5 seconds after the motion stops, when it has decayed to its initial amplitude (e.g., 20%), and the predicted decay ratio. This difference threshold can be set to 15% (i.e., the actual decay ratio is no more than 15 percentage points faster or slower than the prediction). 0.5 seconds after the motion stops is chosen as the observation point because this is the free decay phase of vibration, which clearly reflects the damping characteristics; the 15% difference threshold is an empirical engineering value, and within this range, the model can be considered to have basically accurately captured the vibration damping dynamics.

[0100] When optimization is deemed necessary, iterative adjustments should be made following these principles: Adjusting the reverse judgment: If the actual vibration decays much faster than the predicted vibration (i.e., the actual vibration stops faster), this indicates that the model underestimates the actual damping of the system, and the modal damping ratio of the corresponding modes (usually the first and second bending modes) in the coupled dynamics model should be increased.

[0101] If the actual vibration decays much slower than the predicted vibration (i.e., the actual vibration lasts longer), this indicates that the model overestimates the actual damping of the system, and the modal damping ratio of the corresponding mode in the model should be reduced.

[0102] Determining the adjustment range: The initial adjustment range can be set to 20% to 50% of the current damping ratio. For example, if the current first-order modal damping ratio is 0.02 (2%), and it needs to be increased, it can be tried to adjust it to 0.024 (an increase of 20%) or 0.03 (an increase of 50%). This is a tentative, proportional adjustment strategy that can quickly approach the true value and avoid divergence due to excessively large adjustment steps or slow convergence due to excessively small adjustment steps.

[0103] Based on the above principles, modify the modal damping ratio parameters in the coupled dynamics model. Using the exact same test trajectory instructions, rerun the simulation of the coupled dynamics model to obtain new predicted total joint torque sequences and predicted vibration acceleration sequences. Repeat the above data comparison and judgment process until the torque error and attenuation characteristic differences both meet their respective threshold requirements.

[0104] When the iteration meets the termination condition, the obtained modal damping ratio value is fixed as the calibration parameter for that specific robotic arm and formally updated into the coupled dynamics model. This optimized model will be used for all subsequent feedforward observer design and vibration suppression control of the robotic arm, thereby ensuring the effectiveness and robustness of the control method in practical applications.

[0105] In this embodiment, a series of fixed parameters and initial values ​​characterizing the physical properties and test conditions of the robotic arm are set as the benchmark scenario for the dynamic analysis in this embodiment. Specifically, these include: a first robotic arm (rigid arm) with a length of 0.3m and a mass of 2kg; and a second robotic arm (flexible arm) with a length of 0.5m and a mass of 1.2kg, and its section bending stiffness is set to 120 based on the Euler-Bernoulli beam model. The initial excitation experiment identified the natural frequencies of the first two dominant bending modes as 8.5 Hz and 52 Hz, respectively, with the corresponding mode shapes using the standard cantilever beam function. The test trajectory was planned as a high-speed point-to-point motion from rest to the target point, with a maximum joint angular velocity exceeding [missing information]. The absolute value of the maximum joint angular acceleration is not less than 150. To ensure a fully responsive and flexible dynamics design, the first and second modal damping ratios of the second robotic arm are preset to 0.015. The table below shows a comparison between the actual data collected during a high-speed start-stop test in this parameterized scenario and the predicted data from the coupled dynamics model simulation under different modal damping ratio parameters. Table 1: Comparison of Actual and Predicted Data for Different Modal Damping Ratio Parameters Combining the above table and Figure 2 , Figure 3 This clearly reveals the performance evolution of the coupled dynamics model before and after parameter optimization, and intuitively confirms the necessity and effectiveness of the modal damping ratio iterative calibration method in this embodiment.

[0106] exist Figure 2 In the simulation, the predicted joint moment curves and the actual joint moment curves show a high degree of consistency in their overall trends, reflecting the strong foundation of the rigid-flexible coupling dynamic model constructed in this embodiment for describing the core dynamic behavior of the robotic arm. However, at certain time points, especially near peaks or troughs of highly dynamic changes, the predicted joint moment curves are slightly higher than the actual joint moment curves. This is because the initial values ​​of the damping parameters in the model are inaccurate. The initially set modal damping ratio is too low, meaning that the model underestimates the energy dissipation of the system during high-speed motion. Therefore, in the simulation, in order to drive the model to track the same motion trajectory as the actual one, the model assumes that a slightly larger joint moment than the actual one needs to be applied to overcome the underestimated damping loss. Figure 2 This indicates that even if the model structure is correct, the misalignment of key damping parameters can directly affect the accuracy of the feedforward torque command, which may introduce small torque errors or residual vibrations in actual control.

[0107] exist Figure 3 The evolution trends of the four curves directly reflect the core process and effect of damping ratio optimization. The initial predicted vibration acceleration curve shows a significantly higher amplitude than the actual vibration acceleration curve throughout the decay process, and exhibits the slowest decay rate. This is a typical manifestation of a severely underestimated damping ratio; the model predicts a weak resistance to vibration and slow energy dissipation, resulting in a larger predicted residual vibration amplitude and longer duration. After the first adjustment (setting the modal damping ratio to 0.02), the amplitude of the predicted vibration acceleration curve is reduced, the decay rate is accelerated, and it is closer to the actual curve, indicating that the parameter adjustment direction is correct. Finally, the predicted vibration acceleration curve output by the optimized and calibrated model (modal damping ratio set to 0.025) almost completely overlaps with the actual vibration acceleration curve during the entire free decay phase after motion stops. This signifies that the model's dynamic response characteristics, especially the vibration decay characteristics, have achieved a high degree of consistency with the physical entity.

[0108] According to Table 1 above, Figure 2 and Figure 3 The subtle deviations in joint torque reveal the limitations of relying solely on theoretical or empirically preset damping parameter ratios. Furthermore, the calibration of the vibration decay curve reflects the crucial role of iterative optimization of the modal damping ratio through comparison of measured and simulated data. A model with accurate modal damping ratio parameters ensures that the additional compensation torque calculated by the feedforward observer is precisely measured in both amplitude and phase, thereby achieving effective feedforward cancellation of structural vibrations. This significantly improves the positioning accuracy and stabilization speed of the end effector while ensuring high-speed motion.

[0109] Please see Figure 4 The present invention also provides a robotic arm dynamics analysis and modeling system, which is used to implement the above-mentioned robotic arm dynamics analysis and modeling method, including: A coupled dynamics model module is established to construct a multi-rigid-body dynamics basic model for the first and second robotic arms connected by a rotary joint; and for the second robotic arm, the assumed modal method is used to describe its flexible deformation as a linear superposition of low-order mode shapes to construct a coupled dynamics model. The experimental modal parameter acquisition module is used to obtain the key dynamic parameters of the second robotic arm through excitation experiments, including at least the natural frequency, mode shape function and modal damping ratio, and input them into the coupled dynamic model; The state space equation generation module is used to select a reference time point on a predetermined high-speed motion trajectory, use the desired state at the reference time point as a benchmark, define the state variables as the deviations of each state quantity relative to the desired state, linearize the coupled dynamics model at the reference time point, and derive the linearized state space equation. The compensation torque calculation module is used to design a feedforward observer based on the linearized state-space equation. During the offline simulation stage, the trajectory command of the predetermined high-speed motion trajectory is input into the feedforward observer, and the additional compensation torque used to suppress the residual vibration of the second robotic arm is calculated through the open-loop method. The damping parameter iterative calibration module is used to perform tests including high-speed start and stop on the robotic arm, collect the actual joint torque and the actual vibration acceleration at the end of the second robotic arm; at the same time, the trajectory command of the test is input into the coupled dynamics model for simulation to obtain the corresponding predicted value; the collected actual value is compared with the corresponding predicted value to determine whether the modal damping ratio parameter in the coupled dynamics model needs to be optimized.

[0110] The above formulas are all dimensionless calculations. The formulas are derived from software simulations based on a large amount of collected data to obtain the most recent real-world results. The preset parameters in the formulas are set by those skilled in the art according to the actual situation.

[0111] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented in software, the above embodiments can be implemented, in whole or in part, as a computer program product. Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution.

[0112] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment, depending on actual needs.

[0113] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A method for analyzing and modeling the dynamics of a robotic arm, characterized in that, The specific steps include: Step 1: For the first and second robotic arms connected by a rotary joint, construct their basic multi-rigid-body dynamics model; and for the second robotic arm, use the assumed modal method to describe its flexible deformation as a linear superposition of low-order mode shapes to construct a coupled dynamics model. Step 2: Obtain the key dynamic parameters of the second robotic arm through excitation experiments, including at least the natural frequency, mode shape function, and modal damping ratio, and input them into the coupled dynamic model; Step 3: Select a reference time point on the predetermined high-speed motion trajectory. Using the desired state at the reference time point as the benchmark, define the state variables as the deviations of each state quantity from the desired state. Linearize the coupled dynamics model at the reference time point and derive the linearized state space equation. Step 4: Design a feedforward observer based on linearized state-space equations; In the offline simulation stage, input the trajectory command of the predetermined high-speed motion trajectory into the feedforward observer, and calculate the additional compensation torque for suppressing the residual vibration of the second robotic arm through an open-loop strategy. Step 5: Perform a test on the robotic arm that includes high-speed start and stop, and collect the actual joint torque and the actual vibration acceleration at the end of the second robotic arm; at the same time, input the trajectory command of the test into the coupled dynamics model for simulation and obtain the corresponding predicted value; compare the collected actual value with the corresponding predicted value to determine whether the modal damping ratio parameter in the coupled dynamics model needs to be optimized.

2. The method for analyzing and modeling the dynamics of a robotic arm according to claim 1, characterized in that, Based on the recursive Newton-Euler algorithm, the multi-rigid-body dynamics equations of the rigid running parts of the first and second robotic arms are established to construct the basic model of multi-rigid-body dynamics. The logic for describing the flexible deformation of the second robotic arm is as follows: the second robotic arm is modeled as a cantilever beam with one end fixed and the other end free. Using the hypothetical modal method, the lateral bending elastic deformation of the second robotic arm in the motion plane is described as a linear superposition of low-order mode shapes. Specifically, this includes: expressing the lateral elastic displacement function of the second robotic arm along its length as the sum of the products of each order mode shape function and the corresponding order mode generalized coordinates; wherein, the low-order mode shapes are characterized by the bending frequency and the bending mode shape function; the mode generalized coordinates are independent generalized coordinates describing the change of the vibration amplitude of the corresponding order mode with time; the mode shape function at least includes functions corresponding to the first and second order bending modes of the second robotic arm.

3. The method for analyzing and modeling the dynamics of a robotic arm according to claim 2, characterized in that, By using the second type of Lagrange equations, the multi-rigid-body dynamics equations are energetically coupled with the flexible deformation description to construct a coupled dynamics model, specifically including: Define a complete generalized coordinate vector for the robotic arm, which includes the first joint angle describing the rotation of the first joint, the second joint angle describing the rotation of the second joint, and all modal generalized coordinates describing the second robotic arm; the first joint connects the base and the first robotic arm and drives the first robotic arm to rotate, and the second joint connects the first robotic arm and the second robotic arm and drives the second robotic arm to rotate. The total kinetic energy of the robotic arm is calculated, including the kinetic energy of the first robotic arm, the kinetic energy of the second robotic arm due to the rigid motion of its center of mass, and the additional kinetic energy of the second robotic arm due to its elastic deformation. When calculating the additional kinetic energy, the second robotic arm is integrated along its length, and the kinetic energy component generated by the lateral elastic deformation velocity of each cross section is accumulated. Calculate the total potential energy of the robotic arm, including the gravitational potential energy of the first and second robotic arms, and the elastic potential energy generated by the bending deformation of the second robotic arm; Substituting the total kinetic energy and total potential energy into the second type of Lagrange equations, we obtain a second type of differential equation with the complete generalized coordinate vector as the variable. This second type of differential equation is the coupled dynamic equation; the coupled dynamic model is expressed by this coupled dynamic equation.

4. The method for analyzing and modeling the dynamics of a robotic arm according to claim 3, characterized in that, Obtaining the key dynamic parameters specifically includes: Multiple acceleration sensors are arranged on the body of the second robotic arm, and an excitation is applied to its free end region. The excitation is either a pulse excitation or a broadband random excitation. The vibration response signals of each accelerometer under excitation are collected, and the excitation signal is collected simultaneously. The collected excitation and vibration response signals are processed, and the natural frequencies, mode shape functions, and modal damping ratios of the first two dominant bending modes of the second robotic arm in its motion plane are identified and extracted through frequency response function analysis or operational modal analysis. The extracted modal vibration function is directly assigned to the preset modal vibration function used when constructing the coupled dynamics model; the extracted modal damping ratio is input as a known parameter into the coupled dynamics equation to characterize the internal resistance characteristics of the material and structure of the second robotic arm.

5. The method for analyzing and modeling the dynamics of a robotic arm according to claim 4, characterized in that, For a pre-planned high-speed trajectory that includes high-speed motion segments, the logic for linearizing the coupled dynamics model at the reference time point is as follows: The high-speed motion trajectory is a time series that describes the expected state of each joint of the robotic arm at each time point, including the expected rotation angle, expected angular velocity, and expected angular acceleration. The time point when the second joint angular velocity is the largest or the time point when the second joint angular acceleration reaches its peak value in the high-speed motion trajectory is selected as the reference time point for characterizing the high-speed dynamic characteristics. Obtain the desired rotation angle and desired angular velocity of the first and second joints at the reference time point, and set all modal generalized coordinates and their velocities of the second robotic arm to zero to form the reference state vector corresponding to the reference time point; Substituting the reference state vector into the coupled dynamic equations and setting all generalized accelerations to zero, the joint motor torque required to maintain static equilibrium under this reference state is obtained by solving the equations. This torque is called the static equilibrium torque. The state variables are the actual rotation angle and actual angular velocity of each joint, as well as the actual modal generalized coordinates and actual speed of the second robotic arm. The state variable vector is defined as the deviation between each state variable and the corresponding component in the reference state vector. Define an input variable vector whose elements are the deviations between the actual output torque of each joint motor and the corresponding components in the static balance torque; At the reference time point, using the above deviation as a new variable, a first-order Taylor expansion of the coupled dynamics equation is performed to derive the state matrix and input matrix describing the dynamic evolution of the deviation. Based on the state matrix and the input matrix, a linearized state-space equation is constructed at the reference time point.

6. The method for analyzing and modeling the dynamics of a robotic arm according to claim 1, characterized in that, The design of the feedforward observer specifically includes: Linearized state-space equations are used as forward prediction models to describe the rigid-flexible coupling dynamics of the robotic arm. The model is run in a feedforward open-loop mode, with the trajectory command of the high-speed motion trajectory as the expected input. The trajectory command includes at least the expected angle, expected angular velocity and expected angular acceleration of each joint at each moment, which constitutes the feedforward observer.

7. The method for analyzing and modeling the dynamics of a robotic arm according to claim 1, characterized in that, The feedforward observer recursively calculates the corresponding system state prediction sequence based on the linearized state space equation. The system state prediction sequence includes the predicted state of the flexible deformation of the second robotic arm. The total feedforward torque required to generate the system's predicted state sequence is calculated from the operation of the feedforward observer; the part directly related to suppressing flexible deformation is separated from this total feedforward torque and defined as the additional compensation torque. The additional compensation torque is a time-varying sequence whose amplitude and frequency characteristics are related to the acceleration variation of the high-speed motion trajectory and the natural frequency of the second robotic arm. This additional compensation torque is ultimately superimposed on the torque command of the joint motor to actively counteract the structural vibrations excited by the high-speed motion.

8. The method for analyzing and modeling the dynamics of a robotic arm according to claim 1, characterized in that, The test experiment including high-speed start and stop must meet at least one of the following conditions: the maximum linear velocity of the end effector of the second robotic arm during the movement process is not less than 2 m / s, and the absolute value of the joint angular acceleration within the motion segment of the test trajectory obtained through the test experiment is not less than 100 radians per square second. When controlling the robotic arm to actually execute the test trajectory, the actual joint torque converted from the actual output current of the joint motors driving the first and second robotic arms is collected simultaneously, as well as the actual vibration acceleration measured by the accelerometer installed on the end effector of the second robotic arm.

9. The method for analyzing and modeling the dynamics of a robotic arm according to claim 8, characterized in that, Determining whether to optimize the modal damping ratio specifically includes: The trajectory commands corresponding to the test including high-speed start and stop are input into the coupled dynamics model for numerical simulation, and the predicted total joint torque sequence and predicted vibration acceleration sequence of the simulation output are obtained. The torque error between the actual joint torque and the predicted total joint torque sequence is compared, and the difference in decay characteristics between the actual vibration acceleration and the predicted vibration acceleration sequence after the motion stops is analyzed; the decay characteristics are characterized by the decay rate. If the torque error or the difference in attenuation characteristics exceeds the preset threshold, the modal damping ratio in the coupled dynamics model will be increased or decreased accordingly, depending on whether the attenuation rate of the actual vibration acceleration is faster or slower than the attenuation rate of the predicted vibration acceleration. The updated modal damping ratio is re-introduced into the coupled dynamics model, and the above steps are iteratively executed until the torque error and attenuation characteristic difference do not exceed the preset threshold. The last updated modal damping ratio parameter is then used in the subsequent control of the robotic arm.

10. A system for analyzing and modeling the dynamics of a robotic arm, characterized in that, The robotic arm dynamics analysis and modeling system is used to implement the robotic arm dynamics analysis and modeling method according to any one of claims 1-9, including: A coupled dynamics model module is established to construct a multi-rigid-body dynamics basic model for the first and second robotic arms connected by a rotary joint; and for the second robotic arm, the assumed modal method is used to describe its flexible deformation as a linear superposition of low-order mode shapes to construct a coupled dynamics model. The experimental modal parameter acquisition module is used to obtain the key dynamic parameters of the second robotic arm through excitation experiments, including at least the natural frequency, mode shape function and modal damping ratio, and input them into the coupled dynamic model; The state space equation generation module is used to select a reference time point on a predetermined high-speed motion trajectory, use the desired state at the reference time point as a benchmark, define the state variables as the deviations of each state quantity from the desired state, linearize the coupled dynamics model at the reference time point, and derive the linearized state space equation. The compensation torque calculation module is used to design a feedforward observer based on the linearized state-space equation. During the offline simulation stage, the trajectory command of the predetermined high-speed motion trajectory is input into the feedforward observer, and the additional compensation torque used to suppress the residual vibration of the second robotic arm is calculated through the open-loop method. The damping parameter iterative calibration module is used to perform tests including high-speed start and stop on the robotic arm, collect the actual joint torque and the actual vibration acceleration at the end of the second robotic arm; at the same time, the trajectory command of the test is input into the coupled dynamics model for simulation to obtain the corresponding predicted value; the collected actual value is compared with the corresponding predicted value to determine whether the modal damping ratio parameter in the coupled dynamics model needs to be optimized.

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