Industrial robot tail end positioning compensation method and system considering joint gap error

Abstract

The invention discloses an industrial robot tail end positioning compensation method and system considering a joint gap error, and the method comprises the steps: considering a joint gap of a mechanical arm, and constructing a mechanical arm forward kinematics model containing the gap; the position error data of the tail end of the mechanical arm is actually measured and substituted into the mechanical arm forward kinematics model containing the gap to be solved, the gap data of the tail end of the mechanical arm are obtained, and an optimal distribution model is determined; according to the optimal distribution model, constructing prior probability distribution of gap parameters, calculating a corresponding adaptation probability and a posterior probability, and determining an optimal gap estimation value; and the optimal gap estimation value is substituted into the mechanical arm forward kinematics model containing the gap for joint gap error compensation, and robot tail end positioning compensation is achieved. The method can dynamically adapt to the gap evolution law, reduce the nonlinear error and improve the tail end positioning precision. The industrial robot tail end positioning compensation method and system considering the joint gap error can be widely applied to the technical field of robot positioning.

Description

Technical Field

[0001] This invention relates to the field of robot positioning technology, and in particular to a method and system for end-effector positioning compensation of industrial robots that takes into account joint clearance errors. Background Technology

[0002] Against the backdrop of rapid development in intelligent manufacturing, large robotic arms, with their advantages of heavy load capacity and large working space, are widely used in high-end engineering fields such as aerospace equipment assembly, heavy machinery manufacturing, and large ship welding. With the upgrading of precision machining and high-end equipment manufacturing industries, the market has placed higher demands on the positioning accuracy of the end effector of large robotic arms, and accuracy performance has become the core factor restricting its application expansion.

[0003] During long-term operation, the joint kinematic pairs of large robotic arms experience wear due to repeated friction and alternating loads, resulting in uncertain and random joint clearances. These clearances can induce nonlinear errors in the kinematic transmission chain, severely impacting end-effector positioning accuracy and even reducing operational stability and processing quality consistency, becoming a critical technical bottleneck that urgently needs to be addressed. Existing methods for compensating for robot joint clearance errors have significant limitations: First, they often simplify the clearance to a fixed value in modeling, ignoring its uncertainty and randomness, leading to poor adaptability of the compensation model to actual working conditions and ineffective compensation. Second, research focuses on small and medium-sized robotic arms with arm spans less than 2 meters and loads less than 50 kg, failing to consider the structural size effects and clearance error amplification characteristics of large robotic arms, resulting in significant accuracy degradation when directly applied. Summary of the Invention

[0004] To address the aforementioned technical problems, the present invention aims to provide an end-effector positioning compensation method and system for industrial robots that takes into account joint clearance errors. This method can dynamically adapt to the evolution of clearances, reduce nonlinear errors, and improve end-effector positioning accuracy.

[0005] The first technical solution adopted in this invention is: an end-effector positioning compensation method for industrial robots considering joint clearance errors, comprising the following steps: Considering the joint clearances of the robotic arm, a forward kinematics model of the robotic arm with clearances is constructed by performing quantitative mapping through geometric modeling methods. The measured end-effector position error data was substituted into the forward kinematics model of the robot arm with gaps to obtain the end-effector gap data and determine the optimal distribution model. Construct the prior probability distribution of the gap parameter based on the optimal distribution model, calculate the corresponding fit probability and posterior probability, and determine the optimal gap estimate. The optimal gap estimate is substituted into the forward kinematics model of the robot arm containing the gap to compensate for joint gap error, thereby achieving robot end-effector positioning compensation.

[0006] Furthermore, the step of considering the joint clearances of the robotic arm and constructing a forward kinematic model of the robotic arm containing the clearances through quantization mapping using geometric modeling methods specifically includes: Considering the clearance of the second rotating joint of the robotic arm, an equivalent description is performed using a massless model. The line connecting the center of the robotic arm bushing and the center of the journal is defined as the radial error, and a coordinate system is established to determine the angle between the radial error and the positive axis as the joint error angle. By combining radial error and joint error angle, the system structural error caused by joint clearance is determined, and a single rotary joint error model with clearance is constructed. By performing homogeneous transformation multiplication on the error model of a single rotary joint with clearance, the position and attitude transformation matrix of the end flange surface of a 6-DOF industrial robot relative to the base coordinate system is calculated. Install the target ball to obtain the actual position of the robot arm's end effector, and determine the transformation matrix of the target ball relative to the coordinate system and the transformation matrix of the robot's base coordinate system relative to the laser tracker; Based on the position and attitude transformation matrices of the end flange face of a 6-DOF industrial robot relative to the base coordinate system, the transformation matrix of the target ball relative to the coordinate system, and the transformation matrix of the robot's base coordinate system relative to the laser tracker, a homogeneous transformation matrix is ​​applied along the kinematic chain to determine the positive kinematic relationship between the laser tracker and the target ball at the end. Based on the positive kinematic relationship between the laser tracker and the target ball at the end effector, the position error between the actual position and the ideal position of the end effector is calculated, and a positive kinematic model of the robotic arm with gaps is constructed.

[0007] Furthermore, the expression for the positive kinematic relationship between the laser tracker and the target ball at the end is as follows: ; In the above formula, This represents the transformation matrix of the robot's base coordinate system relative to the laser tracker. The transformation matrix represents the position and orientation of the end flange surface relative to the robot's base coordinate system. This represents the transformation matrix of the target ball relative to the coordinate system. Indicates the angle of twist. Indicates the length of the link. Indicates joint angle, Indicates joint length. Indicates circling Axis rotation, It is expressed as the rotation angle about the x, y, and z axes. It is expressed as the translation distance along the x, y, and z axes. It is expressed as the translation distance along the x, y, and z axes. This indicates the theoretical position of the target ball relative to the laser tracker.

[0008] Furthermore, the step of taking the measured end-effector position error data and substituting it into the forward kinematics model of the robot arm with gaps to obtain the end-effector gap data and determine the optimal distribution model specifically includes: The position error data of the robotic arm's end effector was determined by measuring the end effector position error data using a laser tracker. The end effector structure error data of the robotic arm is substituted into the forward kinematics model of the robotic arm with gaps for nonlinear solution to obtain the end effector gap data of the robotic arm. A pre-defined distribution model is constructed, and the probability density function value is calculated and the natural logarithm is taken based on the gap data of the robotic arm end effector to obtain the log-likelihood values ​​of several distribution models. The distribution model corresponding to the maximum log-likelihood value is selected as the optimal distribution model.

[0009] Furthermore, the step of constructing the prior probability distribution of the gap parameters based on the optimal distribution model, calculating the corresponding fit probability and posterior probability, and determining the optimal gap estimate specifically includes: Construct a Bayesian estimation model and, based on the optimal distribution model, construct the prior probability distribution of the gap parameter; Several sets of candidate gap values ​​are generated based on prior probability distribution sampling; The error propagation model is used to establish the correspondence between several sets of candidate gap values ​​and end position errors. The likelihood function is constructed by combining historical measured position error data to quantify the fitting probability of several sets of candidate gap values. Calculate the posterior probability of several candidate gap values ​​based on a Bayesian estimation model; The optimal gap estimate is determined based on the fit probability of several sets of candidate gap values ​​and the posterior probability of several sets of candidate gap values.

[0010] Furthermore, the expression for the prior probability distribution of the gap parameter is as follows: ; In the above formula, This represents the prior distribution of the gap parameter. This represents the gap error parameter. Represents the Gaussian kernel function. This represents the optimized bandwidth parameters. Indicates the number of samples. Indicates the sample index. Indicates the first The observed values ​​of the gap error parameters corresponding to each sample.

[0011] Furthermore, the step of substituting the optimal gap estimate into the forward kinematics model of the robotic arm containing the gap to compensate for joint gap errors and achieve robot end-effector positioning compensation specifically includes: Substitute the optimal gap estimate into the forward kinematics model of the robotic arm containing the gap to predict the ideal spatial position of the end target ball under the current joint angle combination; Drive the robotic arm to the current joint angle combination and obtain the actual position of the end target ball through a laser tracker; Calculate the distance error between the actual position of the end-effector target ball and the ideal position of the end-effector target ball in space, perform gap compensation verification, and realize robot end-effector positioning compensation.

[0012] The second technical solution adopted in this invention is: an industrial robot end-effector positioning compensation system considering joint clearance error, comprising: The first module is used to consider the joint clearance of the robotic arm, and to construct a forward kinematic model of the robotic arm with clearance by performing quantitative mapping through geometric modeling methods. The second module is used to measure the position error data of the robotic arm end effector and substitute it into the forward kinematics model of the robotic arm with gaps to solve for the gap data of the robotic arm end effector and determine the optimal distribution model. The third module is used to construct the prior probability distribution of the gap parameters based on the optimal distribution model, and to calculate the corresponding fit probability and posterior probability to determine the optimal gap estimate. The fourth module is used to substitute the optimal gap estimate into the forward kinematics model of the robot arm containing the gap to compensate for joint gap error and realize robot end-effector positioning compensation.

[0013] The beneficial effects of the method and system of this invention are as follows: This invention considers the joint clearance of the robotic arm and uses geometric modeling to perform quantization mapping to construct a forward kinematics model of the robotic arm containing clearance. Then, it measures the position error data of the robotic arm's end effector and substitutes it into the forward kinematics model of the robotic arm containing clearance to obtain the clearance data of the robotic arm's end effector and determine the optimal distribution model. It clarifies the quantization and transmission relationship between clearance error, structural error, and position error, and uses the measured position error data to infer clearance information, achieving accurate traceability of clearance parameters. Furthermore, it constructs a prior probability distribution of clearance parameters based on the optimal distribution model and calculates the corresponding fit probability and posterior probability to determine the optimal clearance estimate. Based on a Bayesian framework, it constructs a prior distribution of clearance, and through traversing multiple sets of candidate clearance values ​​and constructing a likelihood function, it fully considers the uncertainty and randomness of clearance. The optimal clearance value is selected for compensation using the maximum posterior probability, dynamically adapting to the clearance evolution law, reducing nonlinear errors, and improving end-effector positioning accuracy. Finally, the optimal clearance estimate is substituted into the forward kinematics model of the robotic arm containing clearance to compensate for joint clearance errors, achieving robot end-effector positioning compensation. Attached Figure Description

[0014] Figure 1 This is a flowchart of the steps of the industrial robot end-effector positioning compensation method that takes into account joint clearance error of the present invention; Figure 2 This is a structural block diagram of the industrial robot end-effector positioning compensation system that takes into account joint clearance error according to the present invention; Figure 3 This is a schematic diagram of a rotary joint model with clearance provided in a specific embodiment of the present invention; Figure 4 This is a schematic diagram of the robot link coordinate system provided in a specific embodiment of the present invention; Figure 5 This is a schematic diagram of the data acquisition and model selection process provided in a specific embodiment of the present invention; Figure 6 This is a schematic diagram of the gap compensation process based on a Bayesian framework provided in a specific embodiment of the present invention; Figure 7 This is a schematic diagram showing the change in the positioning error of the robotic arm end effector before and after considering gap compensation, provided by a specific embodiment of the present invention. Detailed Implementation

[0015] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. The step numbers in the following embodiments are only for ease of explanation and do not limit the order of the steps. The execution order of each step in the embodiments can be adapted according to the understanding of those skilled in the art.

[0016] Reference Figure 1 This invention provides an end-effector positioning compensation method for industrial robots that takes into account joint clearance errors. The method includes the following steps: S100. Considering the joint clearance of the robotic arm, a forward kinematic model of the robotic arm with clearance is constructed by performing quantitative mapping through geometric modeling methods. Specifically, considering the clearance of the second rotary joint of the robotic arm, an equivalent description is performed using a massless model. The line connecting the center of the robotic arm bushing and the center of the journal is defined as the radial error, and a coordinate system is established to determine the angle between the radial error and the positive axis as the joint error angle. Combining the radial error and the joint error angle, the system structural error caused by the joint clearance is determined, and a single rotary joint error model with clearance is constructed. The single rotary joint error model with clearance is multiplied by homogeneous transformation to calculate the position and attitude transformation matrix of the end effector flange of the 6-DOF industrial robot relative to the base coordinate system. A target ball is installed to obtain the actual position of the robotic arm end effector. The transformation matrix of the target ball relative to the coordinate system and the transformation matrix of the robot's base coordinate system relative to the laser tracker are determined. Based on the position and attitude transformation matrices of the end effector flange of the 6-DOF industrial robot relative to the base coordinate system, the transformation matrix of the target ball relative to the coordinate system, and the transformation matrix of the robot's base coordinate system relative to the laser tracker, homogeneous transformation matrices are applied along the kinematic chain to determine the positive kinematic relationship between the laser tracker and the end effector target ball. Based on the positive kinematic relationship between the laser tracker and the end effector target ball, the position error between the actual position and the ideal position of the end effector is calculated, and a positive kinematic model of the robotic arm with gaps is constructed.

[0017] In this embodiment, considering the gap in the second rotating joint of the robotic arm, a geometric modeling method is used to establish a quantitative mapping model between the gap error parameter and the structural parameter error. The robotic arm body, laser tracker and end-effector target ball are regarded as a unified system, and a forward kinematic model of the robotic arm with gap is constructed to clarify the hierarchical transmission relationship of "gap parameter deviation → structural parameter error → end-effector positioning error", forming a complete mathematical link of error transmission, which provides a theoretical basis for subsequent gap back-calculation and compensation.

[0018] First, an error model for a single rotary joint with clearance is established. Considering the clearance in the second rotary joint of the robotic arm, and the random position of the journal within the bushing during robot operation, a massless model is used to equivalently describe the clearance. The clearance equivalent model diagram is shown below. Figure 3 As shown. The gap has been magnified for easier observation.

[0019] Figure 3 middle The center of the bushing is the center of joint rotation when clearance error is not considered. Define the line connecting the journal center and the bushing center as the journal center. This represents the radial error of the joint. Point is the origin. Direction is Establish a coordinate system along the positive axis. Radial error and The included angle in the positive direction of the axis is defined as the joint error angle.

[0020] Without considering joint clearance, the rod The nominal length is The nominal joint angular displacement is Affected by gap error, The actual length becomes The actual joint angular displacement becomes This introduces structural errors. If motion control is still performed according to the nominal structural parameters, the robot's end effector position will deviate from the ideal position. The nominal and actual values ​​of the link length and joint angular displacement satisfy the following: ; According to the above formula, the structural error caused by the joint gap is: ; According to the above formula, the system structural error caused by the joint gap is determined by the radial error and error angle of the joint. Considering the randomness of the joint neck position, according to the joint error model, we know that: ; In the above formula, and These are the radii of the bushing and the journal, respectively.

[0021] According to the above formula, both the radial error and the error angle of the joint are bounded random variables, so the system structural error caused by the joint clearance is a variable function. Since the radial error of the joint is a tiny amount compared to the nominal structural parameters of a large robotic arm, its impact on the end-effector positioning of the robotic arm is negligible compared to the error angle. Therefore, the radial error can be considered as a fixed difference between the bushing and journal radii.

[0022] Furthermore, an end-effector positioning error analysis model is constructed, with the robot link coordinate system as follows: Figure 4 As shown, robot forward kinematics is typically used to determine the coordinate system. arrive Geometric relationship between them: ; in, express function, express function, Indicates the angle of twist. Indicates the length of the link. Indicates joint angle, Indicates joint length. Indicates circling Axis rotation angle, Indicates circling Axis rotation horn, Indicates along Axis translation The distance.

[0023] For an industrial robot with 6 rotary joints, the transformation matrix of the position and orientation of the end flange relative to the robot's base coordinate system can be obtained by multiplying a series of individual homogeneous transformations associated with each joint: ; in, Indicates the spatial attitude of the reference coordinate system. This represents the position vector of the origin of the reference coordinate system.

[0024] During measurement, the actual position information is obtained by installing a target ball at the end of the robotic arm. The transformation matrix of the target ball relative to the coordinate system is as follows: ; in, This indicates location information, in millimeters.

[0025] During measurement, the transformation matrix of the robot's base coordinate system relative to the laser tracker is as follows: ; in, This indicates attitude information, expressed in degrees. This indicates location information, in millimeters.

[0026] Therefore, by applying a homogeneous transformation matrix along the kinematic chain, the positive kinematic relationship between the laser tracker and the target ball at the end can be calculated: ; In the above formula, This represents the transformation matrix of the robot's base coordinate system relative to the laser tracker. The transformation matrix represents the position and orientation of the end flange surface relative to the robot's base coordinate system. This represents the transformation matrix of the target ball relative to the coordinate system. Indicates the angle of twist. Indicates the length of the link. Indicates joint angle, Indicates joint length. Indicates circling Axis rotation, Represented as a small rotation angle about the x, y, and z axes, it describes a small rotation of the robot's base coordinate system relative to the laser tracker. Expressed as the translation distance along the x, y, and z axes, it describes the translational offset of the robot's base coordinate system relative to the laser tracker. Expressed as the translation distance along the x, y, and z axes, it describes the translational offset of the target ball relative to the robot's end flange surface. This indicates the theoretical position of the target ball relative to the laser tracker.

[0027] The presence of joint gaps introduces structural errors into the robot system, resulting in an incorrect actual position of the end effector. Therefore, position error for: ; in, ; In the above formula, The local Jacobian matrix represents the linear mapping between small changes in robot joint parameters and changes in end-effector position. This represents the parameter error vector of joint 2.

[0028] S200, measured end-effector position error data, and substituted into the forward kinematics model of the robot arm with gap to solve for the end-effector gap data and determine the optimal distribution model; Specifically, the end-effector position error data of the robotic arm is measured using a laser tracker to determine the end-effector structural error data; the end-effector structural error data is substituted into the forward kinematics model of the robotic arm with gaps for nonlinear solution to obtain the end-effector gap data; a preset distribution model is constructed, and the probability density function value is calculated and the natural logarithm is taken based on the end-effector gap data to obtain the log-likelihood values ​​of several distribution models; the distribution model corresponding to the maximum log-likelihood value is selected as the optimal distribution model.

[0029] In this embodiment, the position error data of the robotic arm end effector is measured by a laser tracker, and the joint gap error dataset is obtained by back-reasoning based on the error propagation model established in step one. Multiple candidate probability distribution models are constructed, the log-likelihood value of each model for the gap error data is calculated, and the optimal distribution model with the largest log-likelihood value is selected as the statistical description model of the gap parameter, providing a reliable prior distribution basis for Bayesian inference.

[0030] First, based on the end position error, the gap data was inferred. In the actual measurement, N sets of position error data were measured: ; Among them, symbols Indicates the first Group data.

[0031] From N sets of positional error data, N sets of structural error data can be obtained: ; By solving the nonlinear equations, N sets of gap data can be obtained from N sets of structural error data: ; Then, a data distribution model is constructed, and considering the different construction methods of gap data distribution, three different models are introduced.

[0032] 1) Empirical distribution model; The empirical distribution is a "direct mapping" to the sample data. empirical distribution function This represents the percentage of samples less than or equal to x, i.e.: ; in This is an indicator function; it is 1 if the condition is met, and 0 otherwise.

[0033] Since the empirical distribution itself is discrete, a continuous probability density function (PDF) cannot be directly given. However, a continuous "empirical density function" can be constructed through kernel density estimation (KDE) to approximate the true PDF of the population.

[0034] The core of kernel density estimation is to use a "kernel function" to "smoothly diffuse" the probability of each sample point, ultimately superimposing them to obtain a continuous density curve. The empirical probability density function based on the empirical distribution is defined as: ; in, For kernel functions, Gaussian kernels are commonly used. , It's bandwidth. This represents the sample size.

[0035] 2) Normal distribution model; Assuming the data feature x follows a normal distribution, the probability density function of x can be expressed as: ; in: It is the mean, which determines the central location of the distribution; It is variance. The standard deviation determines the degree of dispersion of the distribution.

[0036] 3) Laplace distribution model; Assuming the data feature x follows a Laplace distribution, the probability density function of x can be expressed as: ; in: It is the mean; It is a scale parameter that controls the degree of dispersion of the distribution. It is applicable to symmetrical distributions with many extreme values.

[0037] Then, model selection is performed. When more than one model meets the selection criteria and can reasonably interpret the data, the log-likelihood value can be used to evaluate the reasonableness of various model categories using the data. The log-likelihood value measures the reasonableness of the data under the model; the larger the value, the higher the fit between the model and the data.

[0038] For each data point Calculate its probability density function value under the model. Take the natural logarithm To avoid numerical overflow; summation yields the log-likelihood value: ; Based on gap data and location error data, the three proposed probability models were calculated using the log-likelihood formula. The results in Tables 1 and 2 show that the empirical distribution model performs better than the other two probability models, indicating that this model has a better fit to the data.

[0039] Table 1. Log-likelihood values ​​based on gap data. ; Table 2. Log-likelihood values ​​based on location error data. ; S300. Construct the prior probability distribution of the gap parameters based on the optimal distribution model, and calculate the corresponding fit probability and posterior probability to determine the optimal gap estimate. Specifically, a Bayesian estimation model is constructed, and a prior probability distribution of the gap parameter is built based on the optimal distribution model. Several sets of candidate gap values ​​are generated by sampling based on the prior probability distribution. The correspondence between the several sets of candidate gap values ​​and the end position error is established using the error propagation model, and a likelihood function is constructed by combining historical measured position error data to quantify the fit probability of the several sets of candidate gap values. The posterior probability of the several sets of candidate gap values ​​is calculated based on the Bayesian estimation model. The optimal gap estimate is determined based on the fit probability and the posterior probability of the several sets of candidate gap values.

[0040] In this embodiment, a prior probability distribution of the gap parameters is constructed based on a determined optimal distribution model. When the robotic arm moves to a certain set of joint angle combinations, N sets of candidate gap values ​​are sampled from the prior distribution. The correspondence between the candidate gap values ​​and the end-effector position error is established using an error propagation model. A likelihood function is constructed by combining historical measured position error data to quantify the fit probability of each set of candidate gap values. The posterior probability of each set of candidate gap values ​​is calculated based on Bayes' theorem. The candidate gap value with the highest posterior probability is selected as the optimal gap estimate under that joint angle combination, thereby achieving accurate adaptation to gap uncertainty.

[0041] First, construct the Bayesian estimation model. The classic Bayesian formula can be expressed as: ; in, This is the gap error parameter. This is historical measurement data of the robotic arm's end-effector position error. For the prior distribution of the gap parameter, Let be the likelihood function. The normalization constant is Let be the posterior distribution of the gap parameter.

[0042] Further, a prior distribution is constructed based on samples of gap error parameters. Kernel density estimation is used to establish the empirical distribution. ; Finally, a likelihood function is constructed, which uses the residual data of position errors to build an empirical noise distribution to quantify the matching degree between the candidate gap parameters and historical measured position error data: a theoretical mapping function based on the candidate gap parameters to the end position error. Calculate the residuals between the theoretical position error corresponding to the candidate gap parameter and the N sets of historical measured position errors. The empirical noise distribution is established using kernel density estimation, where the probability density function expression of kernel density estimation is: ; In the above formula, For the amount of residual data, For Gaussian kernel function, The final likelihood function is constructed to obtain the optimized bandwidth parameters. .

[0043] S400: Substitute the optimal gap estimate into the forward kinematics model of the robotic arm containing the gap to perform joint gap error compensation, thereby achieving robot end-effector positioning compensation.

[0044] Specifically, the optimal gap estimate is substituted into the forward kinematics model of the robotic arm containing the gap to predict the ideal spatial position of the end-effector target ball under the current joint angle combination; the robotic arm is driven to move to the current joint angle combination, and the actual position of the end-effector target ball is obtained through a laser tracker; the distance error between the actual position of the end-effector target ball and the ideal spatial position of the end-effector target ball is calculated, gap compensation is verified, and robot end-effector positioning compensation is realized.

[0045] In this embodiment, the optimal gap estimate obtained in step three is substituted into the forward kinematics model of the gap-containing robotic arm to predict the spatial position of the end target; the robotic arm is driven to move to the target position, and the actual position of the end is measured by a laser tracker to calculate the distance error between the predicted position and the measured position; by comparing the change in distance error before and after compensation, the effectiveness of the compensation method is verified, and the positioning accuracy of the end of the large gap-containing robotic arm is finally improved.

[0046] The experimental setup included a large six-degree-of-freedom robotic arm independently developed in the laboratory, a Leica AT960 laser tracker, a computer, and a target sphere. The robot's nominal DH parameters are shown in Table 3. The laser tracker's measurement accuracy can reach [value missing]. It features high precision, high speed, and a large measurement range. When used in conjunction with a target sphere, the laser tracker can perform three-dimensional measurements, providing precise spatial point location information.

[0047] Table 3 DH Parameter Data Table for Robot ; Based on the above conditions, the joint clearance error compensation process is as follows: 1) Randomly generate 60 sets of non-repeating joint angle combinations, calculate the theoretical nominal position of the end target ball based on the nominal kinematic equation of the robotic arm; use a laser tracker to measure the actual spatial position of the end target ball under the corresponding joint angle, calculate the end position error by the difference between the nominal position and the actual position, and then back-calculate the gap error angle data according to the error mapping formula.

[0048] 2) Based on the gap and position error data collected in step 1), construct multiple candidate probability distribution models, calculate the log-likelihood value corresponding to each model, and select the optimal distribution model with the largest log-likelihood value as the statistical descriptive model for the gap parameters and position error data. These two processes are as follows: Figure 5 As shown.

[0049] 3) Randomly select a new set of joint angle combinations. Based on the optimal distribution model determined in step 2), use it as the prior distribution to sample and generate N sets of candidate joint gap error angles. Calculate the theoretical position deviation corresponding to each set of candidate gaps using the kinematic model of the gap-containing robotic arm, and calculate the residual with the historical measured position error. Use the kernel density estimation method to establish the empirical noise distribution of the residuals, and then quantify the likelihood probability of each set of candidate gaps. Combine the prior distribution and the likelihood probability to calculate the posterior probability, and select the candidate gap with the highest posterior probability as the optimal gap estimate under that joint angle.

[0050] 4) Substitute the optimal gap estimate obtained in step 3) into the forward kinematics model of the robotic arm containing the joint gap to complete the prediction of the spatial position of the end target ball under the combination of joint angles.

[0051] 5) Drive the robotic arm to the joint angle combination, measure the actual position of the end target ball using a laser tracker, compare the measured position with the predicted position in step 4), and verify the prediction accuracy of the compensation model.

[0052] 6) Repeat steps 3) to 5) to sequentially complete the joint clearance error compensation and verification for the remaining 60 new spatial positions, ultimately achieving a systematic verification of the joint clearance error compensation effect within the robotic arm's space. Steps 3) to 6) are as follows: Figure 6 As shown. The change in the positioning error of the robotic arm end effector before and after gap compensation is as follows. Figure 7 As shown, the average absolute positioning error at the end of the robotic arm decreased from 1.24 mm to 1.06 mm.

[0053] In summary, this invention, through hierarchical geometric modeling, clarifies the quantitative transmission relationship between gap error, structural error, and positional error. It uses measured positional error data to infer gap information, achieving precise traceability of gap parameters and providing reliable data support for compensation, ensuring the scientific rigor and soundness of the compensation logic. Based on a Bayesian framework, a prior distribution of the gap is constructed. By traversing multiple sets of candidate gap values ​​and constructing a likelihood function (integrating historical residual information), the uncertainty and randomness of the gap are fully considered, overcoming the limitations of traditional fixed-value assumptions. The optimal gap value is selected for compensation using the maximum a posteriori probability, dynamically adapting to the gap evolution law, reducing nonlinear errors, and improving end-effector positioning accuracy. No complex hardware modifications are required; high-precision compensation can be achieved solely through mathematical modeling and Bayesian inference, adapting to the structural characteristics of large robotic arms. It is easy to operate, widely adaptable, and can be quickly implemented in high-end fields such as aerospace assembly and heavy machinery manufacturing, expanding its application boundaries in precision intelligent manufacturing.

[0054] Reference Figure 2 An industrial robot end-effector positioning compensation system that considers joint clearance errors includes: The first module 201 is used to consider the joint clearance of the robotic arm, and to construct a forward kinematic model of the robotic arm with clearance by performing quantitative mapping through geometric modeling methods. The second module 202 is used to measure the position error data of the robotic arm end effector and substitute it into the forward kinematics model of the robotic arm with gap to solve for the gap data of the robotic arm end effector and determine the optimal distribution model. The third module 203 is used to construct the prior probability distribution of the gap parameters based on the optimal distribution model, and to calculate the corresponding fit probability and posterior probability to determine the optimal gap estimate. The fourth module 204 is used to substitute the optimal gap estimate into the forward kinematics model of the robot arm containing the gap to perform joint gap error compensation, thereby realizing robot end-effector positioning compensation.

[0055] The content of the above method embodiments is applicable to this system embodiment. The specific functions implemented in this system embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those achieved in the above method embodiments.

[0056] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.

Claims

1. A method for compensating for end-effector positioning errors in industrial robots, characterized in that, Includes the following steps: Considering the joint clearances of the robotic arm, a forward kinematics model of the robotic arm with clearances is constructed by performing quantitative mapping through geometric modeling methods. The measured end-effector position error data was substituted into the forward kinematics model of the robot arm with gaps to obtain the end-effector gap data and determine the optimal distribution model. Construct the prior probability distribution of the gap parameter based on the optimal distribution model, calculate the corresponding fit probability and posterior probability, and determine the optimal gap estimate. The optimal gap estimate is substituted into the forward kinematics model of the robot arm containing the gap to compensate for joint gap error, thereby achieving robot end-effector positioning compensation.

2. The industrial robot end-effector positioning compensation method considering joint clearance error according to claim 1, characterized in that, The step of considering the joint clearances of the robotic arm and constructing a forward kinematics model of the robotic arm with clearances through quantization mapping using geometric modeling methods specifically includes: Considering the clearance of the second rotating joint of the robotic arm, an equivalent description is performed using a massless model. The line connecting the center of the robotic arm bushing and the center of the journal is defined as the radial error, and a coordinate system is established to determine the angle between the radial error and the positive axis as the joint error angle. By combining radial error and joint error angle, the system structural error caused by joint clearance is determined, and a single rotary joint error model with clearance is constructed. By performing homogeneous transformation multiplication on the error model of a single rotary joint with clearance, the position and attitude transformation matrix of the end flange surface of a 6-DOF industrial robot relative to the base coordinate system is calculated. Install the target ball to obtain the actual position of the robot arm's end effector, and determine the transformation matrix of the target ball relative to the coordinate system and the transformation matrix of the robot's base coordinate system relative to the laser tracker; Based on the position and attitude transformation matrices of the end flange face of a 6-DOF industrial robot relative to the base coordinate system, the transformation matrix of the target ball relative to the coordinate system, and the transformation matrix of the robot's base coordinate system relative to the laser tracker, a homogeneous transformation matrix is ​​applied along the kinematic chain to determine the positive kinematic relationship between the laser tracker and the target ball at the end. Based on the positive kinematic relationship between the laser tracker and the target ball at the end effector, the position error between the actual position and the ideal position of the end effector is calculated, and a positive kinematic model of the robotic arm with gaps is constructed.

3. The industrial robot end-effector positioning compensation method considering joint clearance error according to claim 2, characterized in that, The specific expression for the positive kinematic relationship between the laser tracker and the target ball at the end is as follows: ； In the above formula, This represents the transformation matrix of the robot's base coordinate system relative to the laser tracker. The transformation matrix represents the position and orientation of the end flange surface relative to the robot's base coordinate system. This represents the transformation matrix of the target ball relative to the coordinate system. Indicates the angle of twist. Indicates the length of the link. Indicates joint angle, Indicates joint length. Indicates circling Axis rotation, It is expressed as the rotation angle about the x, y, and z axes. It is expressed as the translation distance along the x, y, and z axes. It is expressed as the translation distance along the x, y, and z axes. This indicates the theoretical position of the target ball relative to the laser tracker.

4. The industrial robot end-effector positioning compensation method considering joint clearance error according to claim 3, characterized in that, The step of obtaining the end-effector position error data by substituting the measured data into the forward kinematics model of the robotic arm with gaps, and then determining the optimal distribution model, specifically includes: The position error data of the robotic arm's end effector was determined by measuring the end effector position error data using a laser tracker. The end effector structure error data of the robotic arm is substituted into the forward kinematics model of the robotic arm with gaps for nonlinear solution to obtain the end effector gap data of the robotic arm. A pre-defined distribution model is constructed, and the probability density function value is calculated and the natural logarithm is taken based on the gap data of the robotic arm end effector to obtain the log-likelihood values ​​of several distribution models. The distribution model corresponding to the maximum log-likelihood value is selected as the optimal distribution model.

5. The industrial robot end-effector positioning compensation method considering joint clearance error according to claim 4, characterized in that, The step of constructing the prior probability distribution of the gap parameters based on the optimal distribution model, calculating the corresponding fit probability and posterior probability, and determining the optimal gap estimate specifically includes: Construct a Bayesian estimation model and, based on the optimal distribution model, construct the prior probability distribution of the gap parameter; Several sets of candidate gap values ​​are generated based on prior probability distribution sampling; The error propagation model is used to establish the correspondence between several sets of candidate gap values ​​and end position errors. The likelihood function is constructed by combining historical measured position error data to quantify the fitting probability of several sets of candidate gap values. Calculate the posterior probability of several candidate gap values ​​based on a Bayesian estimation model; The optimal gap estimate is determined based on the fit probability of several sets of candidate gap values ​​and the posterior probability of several sets of candidate gap values.

6. The industrial robot end-effector positioning compensation method considering joint clearance error according to claim 5, characterized in that, The specific expression for the prior probability distribution of the gap parameter is as follows: ； In the above formula, This represents the prior distribution of the gap parameter. This represents the gap error parameter. Represents the Gaussian kernel function. This represents the optimized bandwidth parameters. Indicates the number of samples. Indicates the sample index. Indicates the first The observed values ​​of the gap error parameters corresponding to each sample.

7. The industrial robot end-effector positioning compensation method considering joint clearance error according to claim 6, characterized in that, The step of substituting the optimal gap estimate into the forward kinematics model of the robotic arm containing the gap to compensate for joint gap error and achieve robot end-effector positioning compensation specifically includes: Substitute the optimal gap estimate into the forward kinematics model of the robotic arm containing the gap to predict the ideal spatial position of the end target ball under the current joint angle combination; Drive the robotic arm to the current joint angle combination and obtain the actual position of the end target ball through a laser tracker; Calculate the distance error between the actual position of the end-effector target ball and the ideal position of the end-effector target ball in space, perform gap compensation verification, and realize robot end-effector positioning compensation.

8. An end-effector positioning compensation system for industrial robots considering joint clearance errors, characterized in that, Includes the following modules: The first module is used to consider the joint clearance of the robotic arm, and to construct a forward kinematic model of the robotic arm with clearance by performing quantitative mapping through geometric modeling methods. The second module is used to measure the position error data of the robotic arm end effector and substitute it into the forward kinematics model of the robotic arm with gaps to solve for the gap data of the robotic arm end effector and determine the optimal distribution model. The third module is used to construct the prior probability distribution of the gap parameters based on the optimal distribution model, and to calculate the corresponding fit probability and posterior probability to determine the optimal gap estimate. The fourth module is used to substitute the optimal gap estimate into the forward kinematics model of the robot arm containing the gap to compensate for joint gap error and realize robot end-effector positioning compensation.

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