Industrial robot multi-joint friction modeling method and related device

Abstract

This application provides a method and related apparatus for multi-joint friction modeling of industrial robots. It collects the angular displacement and current of the joints during the operation of the industrial robot along an excitation trajectory, and obtains the frictional torque by separating the angular displacement, angular velocity, angular acceleration, and current. Weighting coefficients are then assigned to each joint based on the frictional torque. Multiple candidate models are constructed based on angular velocity and angular acceleration. A target model is selected from the candidate models based on their accuracy, complexity, and physical consistency. The target model is then optimized using a loss function to obtain the final model. Weighting coefficients balance the contributions of joints with different frictional torque magnitudes in the loss function. This approach overcomes the trade-off between model discovery capability, computational efficiency, and engineering applicability, providing an accurate, reliable, physically transparent, computationally efficient, and easily deployable solution for industrial robots.

Description

Technical Field

[0001] This application relates to the field of data modeling, and more particularly to a method and apparatus for modeling multi-joint friction of industrial robots. Background Technology

[0002] High-precision control of industrial robots is the core of realizing intelligent manufacturing and precision operation. However, the strong nonlinear frictional torque in the joints severely restricts their trajectory tracking accuracy and motion stability, especially under conditions such as low speed and zero crossing, which can easily cause crawling, oscillation and positioning deviation.

[0003] Current methods for modeling joint friction in industrial robots have the following drawbacks: The model structure depends on prior assumptions; all models require a pre-defined model framework (such as LuGre and Stribeck), lacking the ability to automatically discover better model structures from data, thus limiting the performance ceiling. The dynamic friction characteristics are either poorly characterized or too costly. The model's representation is not conducive to direct integration with the controller; the output is either a differential equation requiring numerical integration or a non-analyzable black / grey box network, neither of which can be easily converted into feedforward compensation terms that are easy to deploy in industrial controllers. The lack of an efficient and unified multi-joint modeling mechanism and the failure to provide a modeling paradigm that can automatically discover the same function form for all joints result in a complex model system and high identification and maintenance costs. Summary of the Invention

[0004] The following is an overview of the topics described in detail in this article.

[0005] The purpose of this application is to at least partially solve one of the technical problems existing in the related technologies. The embodiments of this application provide a method and related apparatus for multi-joint friction modeling of industrial robots.

[0006] An embodiment of the first aspect of this application provides a method for modeling multi-joint friction in an industrial robot, comprising: The angular displacement and current of the joints of the industrial robot during its operation along the excitation trajectory are collected. The angular velocity and angular acceleration of the joints are calculated based on the angular displacement. The frictional torque is obtained by separating the angular displacement, angular velocity, angular acceleration and current. Weighting coefficients are assigned to each joint based on the frictional torque. Based on the angular velocity and the angular acceleration, multiple candidate models are constructed using a genetic programming algorithm. The target model is selected from the candidate models based on their accuracy, complexity, and physical consistency. The target model is optimized based on the loss function to obtain the final model. During the optimization process, the contributions of joints with different friction torque magnitudes to the loss function are balanced by the weighting coefficients.

[0007] According to certain embodiments of the first aspect of this application, the excitation trajectory is a finite-term Fourier series trajectory or a piecewise sine trajectory.

[0008] According to certain embodiments of the first aspect of this application, calculating the angular velocity and angular acceleration of the joint based on the angular displacement includes: The angular displacement is obtained by recording the joint encoder position at each sampling time. The angular velocity and angular acceleration of the joint are obtained by differential calculation of the angular displacement.

[0009] According to certain embodiments of the first aspect of this application, the frictional torque is obtained by separating the angular displacement, the angular velocity, the angular acceleration, and the current, including: Based on the current and torque constants, the estimated torque of the joint is obtained. According to the robot dynamics equations, the frictional torque is obtained by subtracting the motor torques of the forward and reverse motions; The robot dynamics equations are expressed as follows: ; For inertial torque, Coriolis force and centrifugal torque, For gravitational torque, For angular displacement, Angular velocity, Angular acceleration, For frictional torque, To estimate torque for current.

[0010] According to certain embodiments of the first aspect of this application, the weight coefficient of the j-th joint is expressed as: ;in, Let be the weight coefficient of the j-th joint. This represents the total amount of frictional torque data. Let i be the frictional torque of the j-th joint. This is a preset constant.

[0011] According to certain embodiments of the first aspect of this application, the construction of multiple candidate models based on the genetic programming algorithm according to the angular velocity and the angular acceleration includes: Define the search space and set complexity constraints; Based on the search space and the complexity constraints, multiple random expressions are generated according to the angular velocity and the angular acceleration to construct the population as individuals; Calculate the evaluation index of the individuals in the population, and select high-quality individuals based on the evaluation index; Genetic manipulation of high-quality individuals in a population yields a new population. Perform coefficient optimization and population migration operations on the individuals in the new population; Multiple candidate models are obtained from the population individuals that meet the preset iteration conditions.

[0012] According to certain embodiments of the first aspect of this application, before selecting a target model from the candidate models based on the accuracy, complexity, and physical consistency of the candidate models, the method further includes: The candidate models are evaluated on independent validation sets, and the mean squared error is calculated. Candidate models located on the Pareto front are selected based on the mean square error.

[0013] According to certain embodiments of the first aspect of this application, during the optimization process, a multi-starting point initialization strategy and a robust optimizer are employed to adjust the independent coefficients of the target model for each joint.

[0014] According to a second aspect of this application, an electronic device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the industrial robot multi-joint friction modeling method as described in the first aspect of this application.

[0015] According to a third aspect of this application, a computer storage medium stores computer-executable instructions for performing the industrial robot multi-joint friction modeling method as described in the first aspect of this application.

[0016] The above-mentioned scheme has at least the following beneficial effects: By using physically guided symbolic regression technology, it breaks free from the dependence on fixed prior model frameworks, automatically discovering the optimal explicit function structure shared by all joints from the data, thus achieving automatic discovery and unification of model structure. By strategically introducing key features such as angular acceleration and combining them with physically meaningful mathematical operators, the obtained static analytical function can accurately describe the complex characteristics of friction, such as dynamic hysteresis, without relying on differential equations. The final model is a concise explicit analytical expression, possessing both accuracy and generalization ability, and also exhibiting the characteristics of forward computation and no need for iterative integration, which can be directly and efficiently integrated into the real-time control architecture as a feedforward compensation term. Through a multi-joint weighted joint identification strategy, it overcomes the differences in torque magnitude between joints, accurately identifying the parameters of all joints at once under a unified model framework, achieving collaborative and efficient identification of multi-joint systems, and improving overall modeling efficiency. It breaks through the trade-off between model discovery capability, computational efficiency, and engineering applicability, providing industrial robots with a novel friction modeling and compensation solution that is accurate, reliable, physically transparent, computationally efficient, and easy to deploy. Attached Figure Description

[0017] The accompanying drawings are used to provide a further understanding of the technical solutions of this application and constitute a part of the specification. They are used together with the embodiments of this application to explain the technical solutions of this application and do not constitute a limitation on the technical solutions of this application.

[0018] Figure 1 This is a step-by-step diagram of the multi-joint friction modeling method for industrial robots; Figure 2 This is a flowchart illustrating the steps of constructing multiple candidate models based on angular velocity and angular acceleration using a genetic programming algorithm. Figure 3 This is a schematic diagram showing the contributions of the dynamic and static terms of the physics-guided symbolic regression model to joint 1 of the industrial robot. Figure 4 This is a graph showing the characteristics of the coupling variables of the dynamic terms of the physical-guided symbolic regression model at joint 1 of an industrial robot. Figure 5 This is a comparison of the fitting performance of the PGSR model and the mechanism model on joint 1 of an industrial robot; Figure 6 This is a comparison chart of the predictive performance of the PGSR model and the mechanism model on joint 1 of an industrial robot; Figure 7 This is a comparison chart of the fitting performance of the PGSR model and the data-driven model on joint 1 of an industrial robot; Figure 8 This is a comparison chart of the predictive performance of the PGSR model and the data-driven model on joint 1 of an industrial robot. Figure 9 This is a schematic diagram of the velocity and PGSR predicted friction of the PGSR model on joint 4 of an industrial robot. Figure 10 This is a schematic diagram comparing the feedback torque of the PGSR model on joint 4 of an industrial robot; Figure 11 This is a schematic diagram of the prediction error of the friction torque of the PGSR model on joint 4 of an industrial robot. Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0020] It should be noted that although functional modules are divided in the device schematic diagram and a logical order is shown in the flowchart, in some cases, the steps shown or described may be performed in a different order than the module division in the device or the order in the flowchart. The terms "first," "second," etc., in the specification, claims, or the aforementioned drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence.

[0021] First, let's analyze some of the terms used in this application: The Stribeck effect refers to the nonlinear phenomenon where friction gradually decreases with increasing speed. It is common in low-speed motion scenarios and is an important characteristic that needs to be considered in joint friction models.

[0022] LuGre friction model: A dynamic friction model that captures static friction, dynamic friction, and the transition between them by simulating the behavior of microscopic "hairs". It can describe complex friction phenomena such as pre-slip, Stribeck effect, and hysteresis effect.

[0023] Inertia matrix : A symmetric positive definite matrix describing the effect of the mass distribution of a rigid body on rotational motion, reflecting the rotational inertia of the object, and related to its shape and mass distribution.

[0024] Coriolis force and centrifugal force matrix In a rotating coordinate system, the matrix of inertial forces generated by the motion of the coordinate system. The Coriolis force is related to the relative velocity and angular velocity of the object, while the centrifugal force is related to the radius of rotation and angular velocity.

[0025] gravity matrix : This matrix represents the effect of gravity on the joint torque of the robot. It is related to the mass of the robot link and the position of the center of mass, and varies with the joint position.

[0026] Frictional torque Frictional torque of a robot joint is a force that hinders the relative motion of objects, generated by the roughness of the contact surface and intermolecular forces. It is affected by joint speed, load, and surface condition.

[0027] Joint variables: Variables used to describe the state of a robot's joints, such as position, velocity, and acceleration, and are the foundation of robot kinematics and dynamics analysis.

[0028] Symbolic regression is a machine learning technique that aims to discover the optimal mathematical expression between variables to describe the relationship between input and output variables. Unlike traditional linear or multinomial regression, it does not presuppose the structure of the model; instead, it automatically searches for possible mathematical expressions through algorithms to find the model that best fits the data.

[0029] Traditional friction modeling methods mainly include the following three technical paradigms.

[0030] Classical static friction models, such as the Coulomb model, the Coulomb-viscous model, and the Stribeck model, are widely used in engineering practice due to their simple structure and clear physical meaning of parameters. However, they essentially express frictional force as a function of instantaneous velocity, failing to characterize the hysteresis characteristics of the pre-sliding stage and the dynamic frictional hysteresis phenomenon in the macroscopic sliding zone, resulting in a fundamental deficiency in their expressive power.

[0031] Complex dynamic friction models, represented by the LuGre model, Dahl model, and generalized Maxwell sliding (GMS) model, mechanistically describe the frictional behavior from pre-slip to macroscopic slip by introducing internal state variables (such as the mane-average deformation in the LuGre model). However, their differential equation structure leads to complex parameter identification processes, heavily relying on cumbersome nonlinear optimization algorithms. Furthermore, real-time control requires numerical integration of these differential equations, resulting in a heavy computational burden and potential numerical stability issues, thus posing a high barrier to engineering deployment.

[0032] Data-driven models, such as neural networks, possess powerful nonlinear fitting capabilities. However, their "black box" nature results in a complete lack of physical interpretability, making them difficult to embed into industrial control architectures with extremely high reliability and security requirements.

[0033] To address the above issues, embodiments of this application provide a method and related apparatus for modeling multi-joint friction in industrial robots.

[0034] The embodiments of this application will be further described below with reference to the accompanying drawings.

[0035] An automated, physically-guided model discovery and identification system is constructed, which models multi-joint friction data for industrial robots according to the following method. The overall system framework consists of four core modules working in tandem: a data preprocessing and dynamics separation module, a physically-guided symbolic regression module, a model selection module, and a parameter identification and output module. The framework takes the robot's raw motion and current data as input and outputs a unified, explicit, and physically interpretable analytical friction model applicable to all joints.

[0036] Reference Figure 1 A method for modeling friction at multiple joints in industrial robots, comprising the following steps: Step S100: Collect the angular displacement and current of the joints of the industrial robot during its operation along the excitation trajectory; calculate the angular velocity and angular acceleration of the joints based on the angular displacement; obtain the frictional torque by separating the angular displacement, angular velocity, angular acceleration and current; and assign weight coefficients to each joint based on the frictional torque. Step S200: Construct multiple candidate models based on angular velocity and angular acceleration using a genetic programming algorithm; Step S300: Select the target model from the candidate models based on their accuracy, complexity, and physical consistency. Step S400: Optimize the parameters of the target model according to the loss function to obtain the final model. During the optimization process, the contribution of joints with different friction torque magnitudes in the loss function is balanced by weighting coefficients.

[0037] First, in the data preprocessing and dynamics separation module, an excitation trajectory is generated to drive the robot along the specific trajectory and collect data. Then, pure friction torque data is extracted using dynamics separation technology, and a weighted strategy is employed to balance the magnitude differences between multiple joints. The processed data is then fed into a physics-guided symbolic regression module, which automatically searches for candidate mathematical model expressions under preset physical rules (such as the requirement to include acceleration features and the preference for hyperbolic tangent functions). Multiple candidate models generated from the search enter the model selection module, where they are comprehensively scored based on accuracy, simplicity, and physical consistency to select the optimal unified function structure. Finally, in the parameter identification and output module, the coefficients of this unified structure are fine-tuned, outputting the final friction model where all joints share the same function form and only the parameters differ. The entire process achieves automated discovery from data to a physically interpretable analytical model.

[0038] Step S100 is performed by the data preprocessing and dynamics separation module, aiming to obtain a high-quality friction torque dataset that can be used for modeling from the raw sensor data and to solve the problem of inconsistent dimensions of multi-joint data.

[0039] To fully stimulate the nonlinear triboelectric characteristics of the robot's joints, it is necessary to generate excitation trajectories covering low-speed, high-speed, and forward and reverse motions. Finite Fourier series (FFS) trajectories or piecewise sinusoidal trajectories are preferred. In practice, it is essential to ensure smooth trajectory motion and that the angular displacement, angular velocity, and angular acceleration of each joint do not exceed their physical limits.

[0040] Each sampling moment is recorded by the robot controller. Lower joint encoder position This refers to angular displacement; and angular velocity is obtained by differential calculation or observer estimation of the angular displacement. and angular acceleration Simultaneously, the motor current is collected. .

[0041] Using motor current I and torque constant This can be converted into an approximate value of the joint drive torque, i.e. The estimated torque T for the current at each joint of the robot is calculated. This torque includes inertial force, gravity, Coriolis force, and friction. The estimated torque for the current is a part of the motor output torque.

[0042] Based on the robot dynamics equations Frictional torque is separated by subtracting the torques of the motors moving in the forward and reverse directions. ,in Represents the moment of inertia. It consists of Coriolis force and centrifugal torque. It is the gravitational torque. This represents frictional torque. Its physical basis is the torque of inertia. and gravitational torque In two sets of motions with the same path but opposite directions, the frictional force remains constant, while its direction is always opposite to the velocity direction. Control a joint of the robot to perform uniform (or low-speed) forward motion and uniform (or low-speed) reverse motion along identical spatial paths, respectively, and record the joint torques in both sets of motions. and Frictional torque can be achieved through... Approximate separation: This formula, under low-speed conditions, ignores the slight error caused by the asymmetry between the directions of the Coriolis force and the centrifugal force, which is acceptable in engineering.

[0043] To obtain a more accurate frictional torque, after obtaining the preliminary dynamic parameters, a compensation calculation can be performed using a complete dynamic model to obtain the frictional torque. Among them, the inertial torque and gravitational torque Coriolis torque These parameters can be obtained in advance using standard dynamic parameter identification methods (such as weighted least squares method).

[0044] Because the loads and transmission ratios of the robot's joints differ, their frictional torques can vary by tens of times. Direct joint training can lead to the loss function being dominated by high-torque joints, while the parameters of low-torque joints are poorly identified.

[0045] Therefore, a statistically based automatic weighting strategy is designed. This strategy applies to each joint. Assign a weighting factor, which is expressed as: , among which, among which, Let be the weight coefficient of the j-th joint. This represents the total amount of frictional torque data. Let i be the frictional torque of the j-th joint. This is a preset constant. The purpose of this weight is to give higher weights to joints with smaller overall frictional torque in the loss function, thus balancing their contribution. Selected This strategy effectively balances the influence of each joint without excessively suppressing high-level information, ensuring the accuracy of the final identified joint independence coefficients. It has real physical meaning and correct dimensions, ensuring the physical consistency of the model.

[0046] To address the issue of inaccurate joint parameter identification caused by the large differences in the magnitude of frictional torques among robot joints, this strategy defines a weighting coefficient for each joint based on the statistical characteristics of its frictional torque data (such as powers of the sum of magnitudes). This weights the error term of the joint in the training loss function, thereby balancing the contributions of all joints within a unified optimization framework and achieving synchronous and accurate identification of all joint parameters.

[0047] For step S200, execution is performed via the physically guided symbolic regression module. Joint angular velocity is selected. and angular acceleration As input, the core physical basis lies in the historical dependence of the hysteresis characteristics of friction, which is closely related to the rate of change of the motion state. Symbolic regression is implemented based on a genetic programming algorithm.

[0048] Reference Figure 2 Based on the genetic programming algorithm, multiple candidate models are constructed according to angular velocity and angular acceleration, including the following steps: Step S210: Define the search space and set complexity constraints; Step S220: Based on the search space and complexity constraints, generate multiple random expressions as individuals in the population to construct the population according to angular velocity and angular acceleration; Step S230: Calculate the evaluation index of individuals in the population, and select high-quality individuals based on the evaluation index. Step S240: Perform genetic operations on high-quality individuals in the population to obtain a new population; Step S250: Perform coefficient optimization and population migration operations on the individuals in the new population. Step S260: Multiple candidate models are obtained from the population individuals that meet the preset iteration conditions.

[0049] Define the search space, which is the "set of elements" that can be used in the expression. The core is to determine candidate components by combining physical priors: it includes physical symbols (such as physical quantities such as time t, displacement s, force F, etc.); it includes basic mathematical / physical functions (such as power functions, trigonometric functions, exponential functions, etc.); it includes operational operators (such as +, -, ×, ÷) and constant terms.

[0050] Set complexity constraints. Limit the "complexity" of expressions using quantifiable metrics to ensure the physical interpretability of subsequently generated expressions. Define the quantification method for complexity: such as "number of terms," ​​"number of nested functions," or "number of parameters"; set constraint thresholds: such as limiting the number of terms to ≤5, the number of nested functions to ≤2, and the number of parameters to ≤4, to avoid overly redundant expressions.

[0051] Based on search space and complexity constraints, multiple initial candidate expressions are generated, i.e., individuals in the population. Symbols, functions, and operators are randomly selected from the search space for combination; ensuring that each generated expression satisfies complexity constraints (e.g., the number of terms does not exceed 5); finally, a population of a certain size (e.g., 100 different initial expressions) is generated as the starting point for modeling.

[0052] In this embodiment, during the process of automatically searching for mathematical model structures from data using a physics-guided symbolic regression algorithm, prior physical knowledge is systematically incorporated as guidance and constraints. Key input features are selected based on friction mechanisms (e.g., angular acceleration is introduced); specific mathematical operators are preferred or excluded based on physical rationality and numerical stability (e.g., preference for tanh, exclusion of sin, inv, etc.). The goal is to ensure that the automatically discovered model possesses both data fitting capabilities and explicit physical interpretability.

[0053] Operators with clear physical meaning, such as the hyperbolic tangent function (tanh), should be preferred, while operators with singularities (such as reciprocals or logarithms) or periodicities (such as sine or cosine) should be avoided. The tanh function can smoothly describe the transition from static to kinetic friction. It has flexible adjustment capabilities: coefficient The slope of the dominant function from the linear region to the saturation region, and the coefficients Controlling the saturation amplitude. By using a mathematical expression that includes a composite variable of velocity and acceleration (such as tanh(c2×acceleration + c3×velocity)), the dynamic hysteresis effect of friction can be effectively characterized without internal differential equations, thus realizing the description of complex dynamic behavior with a concise explicit function.

[0054] For each individual expression in the current population, calculate the core metrics and select high-quality individuals. Predict the error in the physical data using the expression (e.g., mean squared error, MSE); the smaller the loss, the better the fit. Calculate the complexity value for each expression. Update the Pareto front and select the set of individuals that are "more efficient in terms of complexity when the loss is not worse, and more efficient in terms of loss when the complexity is not worse"—that is, high-quality individuals that cannot improve one metric without sacrificing another.

[0055] Genetic operations such as selection, crossover, and mutation are performed on high-quality individuals in the current population to obtain a new population. Selection involves choosing high-quality individuals from the current population (prioritizing those at the Pareto front), for example, selecting the top 30% of individuals based on loss ranking. Crossover involves swapping parts of the structure of two selected individuals (expressions), for example, combining the first half of expression A (e.g., at+) with the second half of expression B (e.g., bsin(t)) to generate a new expression at+bsin(t) (complexity constraints must be met). Mutation involves randomly modifying the local structure of a single individual, for example, replacing sin(t) in an expression with cos(t) in the search space, or adjusting the number of terms (complexity constraints must be met).

[0056] To improve fitting accuracy, the coefficients of individuals in the new population are optimized. For example, BFGS numerical optimization and simulated annealing numerical optimization can be used.

[0057] Perform population migration operations on individuals in the new population. Maintain population diversity and quality, and avoid premature homogenization. Adjust complexity constraints based on the current Pareto front; for example, if the complexity of frontier individuals is generally too high, appropriately reduce the constraints (e.g., reduce the number of terms from 5 to 4) to achieve adaptive complexity scaling. Run multiple independent populations simultaneously, allowing each population to evolve independently, avoiding the limitations of a single population. Every few generations, migrate high-quality individuals from the Pareto front of one population to other populations, allowing the quality coefficient to diffuse among populations.

[0058] Once the population has evolved to the preset number of generations (or the target metric has been met), candidate models and corresponding metrics are output. Furthermore, the genetic algorithm controls the maximum number of nodes in the symbolic regression process to 22, the number of iterations to 22, and the population size to 35, ensuring that the search fully retrieves suitable function expressions.

[0059] This approach automatically identifies the same mathematical function form as the friction model for all joints of a robot (e.g., six joints), serving as a unified friction model structure. The models for each joint differ only in the coefficients (parameters) within the function. This solves the problem of inconsistent model forms and complex maintenance caused by traditional methods of modeling each joint independently, and is key to achieving efficient and concise modeling of multi-joint systems.

[0060] For step S300, the candidate models are evaluated on an independent validation set, and the mean squared error is calculated. Candidate models located on the Pareto front are selected based on the mean squared error. The target model is then selected from the candidate models based on their accuracy, complexity, and physical consistency.

[0061] Specifically, during the model selection phase, all candidate models are evaluated on an independent validation set (data not used in training), and their mean squared error (MSE) is calculated. Models located on the Pareto front are selected, that is, a subset of models that cannot reduce error without increasing complexity.

[0062] The initial screening models were then quantitatively ranked, taking into account accuracy, complexity, and physical consistency. Among these, the physical consistency score... The rewards will be given to operators that contain clearly defined physical meaning, such as... Corresponding to the Coulomb friction model, the linear term Corresponding to viscous friction models, etc.

[0063] For each candidate model Its overall utility score The calculation formula is: ; in, , and These represent the validation set error (reflecting accuracy), complexity, and physical consistency score of the i-th model, respectively. A series of symbolic simplification rules (such as algebraic identity simplification, redundancy removal operators, and constant merging) are applied to the finally selected expression to obtain a unified function structure for subsequent coefficient identification, i.e., the target model.

[0064] After symbolic regression generates numerous candidate mathematical expressions, a two-stage process is used to automatically select the optimal model. This framework not only considers the model's prediction accuracy on the validation set (such as mean squared error MSE), but also quantitatively evaluates the model's simplicity (expression complexity) and physical consistency (scoring according to preset rules). Finally, the optimal model is selected through a weighted comprehensive score, achieving the best trade-off between accuracy, simplicity, and physical rationality.

[0065] For step S400, the target model is optimized according to the loss function to obtain the final model. During the optimization process, the contributions of joints with different friction torque magnitudes to the loss function are balanced by weighting coefficients. Furthermore, a multi-starting-point initialization strategy and a robust optimizer are employed during the optimization process to adjust the independent coefficients of the target model for each joint.

[0066] This paper presents a parameter identification method designed to address the significant differences in the magnitude of frictional torques across multiple joints. The strategy assigns a weight related to the statistical characteristics of the torque to the loss function term of each joint, balancing the contributions of different joints during joint optimization. This allows for accurate and stable identification of specific parameters for all joints in a single training iteration.

[0067] Specifically, a unified function structure is obtained. Subsequently, a multi-starting-point initialization strategy and a robust optimizer based on the Levenberg-Marquardt algorithm were employed to optimize the specific coefficients of each joint. Fine-tuning was performed. Huber loss was used during optimization to suppress the influence of outliers, and L2 regularization and early stopping mechanisms were introduced to prevent overfitting. After final optimization convergence, a friction model for the six joints was obtained. .

[0068] In this embodiment, a physical rule-guided symbolic regression technique is used to overcome the dependence on a fixed prior model framework. It automatically discovers the optimal explicit function structure shared by all joints from the data, achieving automatic discovery and unification of the model structure. By strategically introducing key features such as angular acceleration and combining them with physically meaningful mathematical operators, the obtained static analytical function can accurately describe the complex characteristics of friction, such as dynamic hysteresis, without relying on differential equations. This allows for efficient characterization of dynamic friction using static analytical functions. The final model is a concise explicit analytical expression, combining high precision and high engineering practicality. It possesses accuracy and generalization ability comparable to complex mechanism models, while also exhibiting forward computation and the characteristic of not requiring iterative integration. It can be directly and efficiently integrated into real-time control architectures as feedforward compensation terms. Through a matching multi-joint weighted joint identification strategy, the differences in torque magnitude between joints are overcome. Under a unified model framework, the parameters of all joints are accurately identified in one go, achieving collaborative and efficient identification of multi-joint systems and improving overall modeling efficiency. Breaking through the existing technological dilemma of balancing model discovery capability, computational efficiency, and engineering applicability, this paper provides a novel friction modeling and compensation solution for industrial robots that is accurate, reliable, physically transparent, computationally efficient, and easy to deploy.

[0069] Through the above process, models of the form shown below can often be automatically discovered: .

[0070] The output friction model consists of dynamic and static terms, where the dynamic terms... and static items In static items, It has a high gradient at zero, enabling it to simulate rapid changes from pre-slip to the sliding region, and saturates in the high-speed region; this nonlinear term is similar to the viscous term. The superposition of these factors together forms the complete static friction curve. The introduction of the acceleration term in the model provides it with the ability to identify motion trends, particularly when the system is in the acceleration phase (…). During the deceleration phase, the frictional force changes along the lower branch of the hysteresis ring; while during the deceleration phase ( Frictional force varies along the upper branch; coefficient The negative value ensures physical consistency; positive acceleration corresponds to the trend of frictional force developing in the negative direction, accurately reproducing the frictional hysteresis phenomenon. Further analysis of the composite variables in the dynamic term... It implements adaptive characteristic description under different operating conditions, when Very small, this item is by Dominate, focus on capturing the lag effect; when Larger, by The model is dominated by and returns to the classical velocity-friction relationship. In addition, compared with differential equation models that require complex numerical processing, this model has natural numerical stability. This characteristic stems from the inherent boundedness of the tanh function, whose range is strictly limited to the range of (-1, 1). This naturally constrains the output of the dynamic and nonlinear static terms of the model within a finite interval, fundamentally avoiding the risk of numerical overflow and ensuring the numerical robustness of the simulation and control process.

[0071] Taking a six-joint industrial robot as an example, low-speed and high-speed excitation trajectories are used to collect data from each joint; a unified final model, namely the PGSR model, is obtained through the PGSR framework.

[0072] We evaluate the fitting performance, generalization ability, and computational efficiency by comparing the model with the Stribeck model, LuGre model, and multilayer perceptron (MLP).

[0073] Figure 3 This is a schematic diagram showing the contributions of the dynamic and static terms of the physics-guided symbolic regression model to joint 1 of the industrial robot. Figure 4 This is a graph showing the characteristics of the coupling variables of the dynamic terms of the physical-guided symbolic regression model at joint 1 of an industrial robot. Figure 5 This is a comparison of the fitting performance of the PGSR model and the mechanism model on joint 1 of an industrial robot; Figure 6 This is a comparison chart of the predictive performance of the PGSR model and the mechanism model on joint 1 of an industrial robot; Figure 7 This is a comparison chart of the fitting performance of the PGSR model and the data-driven model on joint 1 of an industrial robot; Figure 8 This is a comparison chart of the predictive performance of the PGSR model and the data-driven model on joint 1 of an industrial robot. Figure 9 This is a schematic diagram of the velocity and PGSR predicted friction of the PGSR model on joint 4 of an industrial robot. Figure 10 This is a schematic diagram comparing the feedback torque of the PGSR model on joint 4 of an industrial robot; Figure 11 This is a schematic diagram illustrating the prediction error of the frictional torque on joint 4 of an industrial robot using the PGSR model. According to... Figures 3 to 11 It can be seen that the PGSR model is comparable to the LuGre model in terms of accuracy, but has higher computational efficiency; compared with MLP, the PGSR model has stronger generalization ability and does not exhibit overfitting; in feedforward compensation control simulation, the PGSR model can effectively improve trajectory tracking accuracy.

[0074] This method can automatically discover, offline, an explicit analytical frictional mathematical model shared by all joints using motion and torque data generated by a robot running on a specific excitation trajectory. The obtained model can be directly used for feedforward compensation in the control system to improve the robot's motion accuracy and stability in tasks such as precision assembly and trajectory tracking.

[0075] Embodiments of this application provide an electronic device. The electronic device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the multi-joint friction modeling method for industrial robots as described above.

[0076] This electronic device can be any smart terminal, including computers.

[0077] In general, for the hardware structure of electronic devices, the processor can be implemented using a general-purpose CPU (Central Processing Unit), microprocessor, application-specific integrated circuit (ASIC), or one or more integrated circuits, to execute relevant programs and implement the technical solutions provided in the embodiments of this application.

[0078] The memory can be implemented in the form of read-only memory (ROM), static storage device, dynamic storage device, or random access memory (RAM). The memory can store the operating system and other applications. When the technical solutions provided in the embodiments of this specification are implemented through software or firmware, the relevant program code is stored in the memory and is called and executed by the processor.

[0079] Input / output interfaces are used to implement information input and output.

[0080] The communication interface is used to enable communication and interaction between this device and other devices. Communication can be achieved through wired means (such as USB, Ethernet cable, etc.) or wireless means (such as mobile network, WIFI, Bluetooth, etc.).

[0081] The bus transmits information between various components of a device, such as the processor, memory, input / output interfaces, and communication interfaces. The processor, memory, input / output interfaces, and communication interfaces communicate with each other within the device via the bus.

[0082] Embodiments of this application provide a computer storage medium. The computer storage medium stores computer-executable instructions for executing the industrial robot multi-joint friction modeling method described above.

[0083] It will be understood by those skilled in the art that all or some of the steps and systems in the methods disclosed above can be implemented as software, firmware, hardware, and suitable combinations thereof. Some or all of the physical components can be implemented as software executed by a processor, such as a central processing unit, digital signal processor, or microprocessor, or as hardware, or as an integrated circuit, such as an application-specific integrated circuit. Such software can be distributed on a computer-readable medium, which can include computer storage media (or non-transitory media) and communication media (or transient media). As is known to those skilled in the art, the term computer storage media includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storing information (such as computer-readable instructions, data structures, program modules, or other data). Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technologies, CD-ROM, digital versatile disc (DVD) or other optical disc storage, magnetic cartridges, magnetic tape, disk storage or other magnetic storage devices, or any other medium that can be used to store desired information and is accessible to a computer. Furthermore, it is well known to those skilled in the art that communication media typically contain computer-readable instructions, data structures, program modules, or other data in modulated data signals such as carrier waves or other transmission mechanisms, and may include any information delivery medium. In the foregoing description of this specification, references to terms such as "one embodiment," "another embodiment," or "some embodiments," etc., indicate that a specific feature, structure, material, or characteristic described in connection with an embodiment or example is included in at least one embodiment or example of this application. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0084] Those skilled in the art will understand that all or some of the steps in the methods disclosed above, as well as the functional modules / units in the systems and devices, can be implemented as software, firmware, hardware, or suitable combinations thereof.

[0085] The units described above as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, 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 according to actual needs.

[0086] Furthermore, the functional units in the various embodiments of this application can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0087] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes multiple instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of this application. The aforementioned storage medium includes various media capable of storing programs, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0088] In the several embodiments provided in this application, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of the units described above is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed between each other may be through some interfaces; the indirect coupling or communication connection between apparatuses or units may be electrical, mechanical, or other forms. Although embodiments of this application have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of this application, the scope of which is defined by the claims and their equivalents.

[0089] The above is a detailed description of the preferred embodiments of this application, but this application is not limited to the embodiments. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of this application, and these equivalent modifications or substitutions are all included within the scope defined by the claims of this application.

Claims

1. A method for modeling multi-joint friction in industrial robots, characterized in that, include: The angular displacement and current of the joints of the industrial robot during its operation along the excitation trajectory are collected. The angular velocity and angular acceleration of the joints are calculated based on the angular displacement. The frictional torque is obtained by separating the angular displacement, angular velocity, angular acceleration and current. Weighting coefficients are assigned to each joint based on the frictional torque. Based on the angular velocity and the angular acceleration, multiple candidate models are constructed using a genetic programming algorithm. The target model is selected from the candidate models based on their accuracy, complexity, and physical consistency. The target model is optimized based on the loss function to obtain the final model. During the optimization process, the contributions of joints with different friction torque magnitudes to the loss function are balanced by the weighting coefficients.

2. The industrial robot multi-joint friction modeling method according to claim 1, characterized in that, The excitation trajectory is a finite-term Fourier series trajectory or a piecewise sine trajectory.

3. The industrial robot multi-joint friction modeling method according to claim 1, characterized in that, The calculation of the angular velocity and angular acceleration of the joint based on the angular displacement includes: The angular displacement is obtained by recording the joint encoder position at each sampling time. The angular velocity and angular acceleration of the joint are obtained by differential calculation of the angular displacement.

4. The industrial robot multi-joint friction modeling method according to claim 1, characterized in that, The frictional torque is obtained by separating the angular displacement, angular velocity, angular acceleration, and current, including: Based on the current and torque constants, the estimated torque of the joint is obtained. According to the robot dynamics equations, the frictional torque is obtained by subtracting the motor torques of the forward and reverse motions; The robot dynamics equations are expressed as follows: ; For inertial torque, Coriolis force and centrifugal torque, For gravitational torque, For angular displacement, Angular velocity, Angular acceleration, For frictional torque, To estimate torque for current.

5. The method for modeling multi-joint friction of industrial robots according to claim 1, characterized in that, The weight coefficient of the j-th joint is expressed as: ;in, Let be the weight coefficient of the j-th joint. This represents the total amount of frictional torque data. Let i be the frictional torque of the j-th joint. This is a preset constant.

6. The method for modeling multi-joint friction of industrial robots according to claim 1, characterized in that, The genetic programming algorithm constructs multiple candidate models based on the angular velocity and the angular acceleration, including: Define the search space and set complexity constraints; Based on the search space and the complexity constraints, multiple random expressions are generated according to the angular velocity and the angular acceleration to construct the population as individuals; Calculate the evaluation index of the individuals in the population, and select high-quality individuals based on the evaluation index; Genetic manipulation of high-quality individuals in a population yields a new population. Perform coefficient optimization and population migration operations on the individuals in the new population; Multiple candidate models are obtained from the population individuals that meet the preset iteration conditions.

7. The method for modeling multi-joint friction of industrial robots according to claim 1, characterized in that, Before selecting a target model from the candidate models based on their accuracy, complexity, and physical consistency, the method further includes: The candidate models are evaluated on independent validation sets, and the mean squared error is calculated. Candidate models located on the Pareto front are selected based on the mean square error.

8. The method for modeling multi-joint friction of industrial robots according to claim 1, characterized in that, During the optimization process, a multi-starting point initialization strategy and a robust optimizer are adopted to adjust the independent coefficients of the target model for each joint.

9. An electronic device, characterized in that, include: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the industrial robot multi-joint friction modeling method as described in any one of claims 1 to 8.

10. A computer storage medium, characterized in that, The system stores computer-executable instructions for performing the industrial robot multi-joint friction modeling method as described in any one of claims 1 to 8.

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