A method for eliminating redundant parameters in a robot exponential product model

Abstract

The present invention discloses a method for eliminating redundant parameters in a robot exponential product model, which includes the following steps: Step 1, establish a robot kinematic model; Step 2, establish a robot error calibration model; Step 3, calculate the Jacobian matrix according to the robot joint variable measurement data set and the robot error calibration model; Step 4, find the columns in the Jacobian matrix that are all zero, and eliminate the corresponding ineffective parameters; Step 5, calculate the correlation coefficients between any two columns in the Jacobian matrix, and eliminate the pairwise correlated parameters according to the calculation results; Step 6, calculate the multiple correlation coefficients between each column in the Jacobian matrix and other columns, and eliminate the multiple correlation coefficients according to the calculation results; Step 7, determine whether all multiple correlation coefficients R≠1, if not, repeat the previous step, if so, the elimination is completed. The present invention can increase the robustness in the process of calibrating robot kinematic parameters, improve the positioning accuracy of the robot, reduce the calculation time of the calibration program, and improve the calibration efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of robots, and is specifically applied to the elimination of redundant parameters in robot kinematic parameter calibration. Background Art

[0002] With the development of technology, robots are widely used in modern industrial production. However, due to the influence of part processing and manufacturing errors, flexible deformation errors of links and joints, transmission errors, etc., the positioning accuracy of robots is relatively low, which limits the application and development of robots in high-precision machining. According to research, about 90% of the positioning errors of industrial robots are caused by structural parameter errors. Therefore, how to obtain accurate robot kinematic parameters is crucial for improving its accuracy, and the method of robot kinematic parameter calibration can solve this problem.

[0003] Robot kinematic parameter calibration includes four steps: modeling, measurement, identification, and compensation. Modeling is the basis of calibration. For the research on robot kinematic parameter calibration, an effective kinematic model must have three characteristics: continuity, completeness, and non-redundancy. Currently, robots mainly use the DH (Denavit-Hartenberg) model, product of exponentials model, CPC (complete and parametrically continuous) model, S (stone) model, etc. for kinematic modeling. However, when calibrating error parameters, there are redundant parameters in the above kinematic models. When calibrating error parameters, there is a linear relationship between redundant parameters and parameters to be identified, which makes the Jacobian matrix singular and directly affects the calibration result. The methods for eliminating redundant parameters in the model include the analytical method and the numerical method. The analytical method directly derives the explicit expression of error parameters, obtains the relationship between error parameters, and eliminates redundant parameters. The numerical method processes the Jacobian matrix to eliminate redundant parameters, such as the QR (Orthogonal-triangular decomposition) decomposition method and singular value decomposition method proposed by W. KHALIL. However, the elimination process is relatively complex for both the analytical method and the numerical method, and all redundant parameters in the product of exponentials model cannot be eliminated. Regarding the redundant parameter problem in the product of exponentials model, He Ruibo, Wu Yuanqing, and others have conducted research, but all are based on the error calibration model under full pose measurement. When using the error calibration model based on position measurement or distance measurement, there are still redundant parameters in the product of exponentials model. In view of the above problems, the present invention proposes a method for eliminating redundant parameters in the robot product of exponentials model, which can eliminate all redundant parameters in the robot product of exponentials model. Summary of the Invention

[0004] The object of the present invention is to propose a method for eliminating redundant parameters in the exponential product model of a robot, which can improve the robustness of the robot kinematic parameter calibration system, improve the calibration effect, and improve the robot positioning accuracy by eliminating redundant parameters.

[0005] To achieve the above object of the invention, the technical solution of the present invention is: a method for eliminating redundant parameters in the exponential product model of a robot, which is used for eliminating redundant parameters in robot kinematic parameter calibration, and specifically includes the following steps:

[0006] Step 1, establish a robot kinematic model;

[0007] Step 2, establish a robot error calibration model;

[0008] Step 3, calculate the Jacobian matrix according to the robot joint variable measurement data set and the robot calibration error model

[0009]

[0010] where n is the number of robot kinematic parameters, p is the number of data in the robot joint variable measurement data set, and j a,b represents the sub-matrix of the Jacobian matrix calculated by differentiating the i-th kinematic parameter according to the a-th group of data, and J i represents the i-th column of the Jacobian matrix, which corresponds to the i-th kinematic parameter of the robot;

[0011] Step 4, find the columns in the Jacobian matrix J that are all zeros, update the Jacobian matrix after eliminating the corresponding ineffective parameters;

[0012] Step 5, calculate the correlation coefficient between any two columns in the Jacobian matrix J

[0013]

[0014] where J i and J j respectively represent the i-th column and the j-th column data of the Jacobian matrix J, and row(J) represents the number of rows of the Jacobian matrix J.

[0015] When r = ±1, it means that there is a linear relationship between J i and J j , and the kinematic parameters corresponding to J i and J j are pairwise correlated parameters. For pairwise correlated parameters, only one of them is retained as an independent parameter, and the other parameters are eliminated, and the Jacobian matrix is updated;

[0016] Step 6, calculate a column J i in the Jacobian matrix J column by column and other columns J1, J2,..., J i-1 , Ji+1 ,…,J n The multiple correlation coefficient R between them is calculated as follows:

[0017] First, use J i and J1, J2, …, J i-1 , J i+1 , …, J n to perform multiple linear regression, and obtain:

[0018]

[0019] It should be noted that when performing multiple correlation analysis on the Jacobian matrix, the existence of a constant term is not allowed. Therefore, the constant term is removed here for multiple linear regression:

[0020]

[0021] Then calculate the correlation coefficient between J i and . This correlation coefficient is the multiple correlation coefficient between J i and J1, J2, …, J i-1 , J i+1 , …, J n . The calculation formula for the multiple correlation coefficient is:

[0022]

[0023] where represents the arithmetic mean of all the data of J i .

[0024] The value range of the multiple correlation coefficient R is [0, 1]. The larger R is, the higher the degree of correlation between J i and J1, J2, …, J i-1 , J i+1 , …, J n . When R = 1, it means that J i and J1, J2, …, J i-1 , J i+1 , …, J n have a perfect linear relationship, and the kinematic parameters corresponding to J i are called multiple correlation parameters, and the Jacobian matrix is updated after removing this parameter;

[0025] Step 7: Determine whether all multiple correlation coefficients R ≠ 1. If not, repeat Step 6; if so, end.

[0026] The beneficial effects of the present invention are:

[0027] The proposed method can eliminate all redundant parameters in the robot exponential product model. By analyzing the correlation relationships among the robot kinematic parameters, all redundant parameters in the robot model are eliminated. After eliminating the redundant parameters, the robustness of the calibration system is significantly improved, and the positioning accuracy of the robot end is effectively enhanced. Meanwhile, the elimination of redundant parameters also reduces the ineffective calculation steps in the calibration program, decreases the iteration times of the calibration program, significantly improves the running speed, and effectively enhances the calibration efficiency. Description of the Drawings

[0028] Figure 1 It is the structure and dimension diagram of the UR5 robot used in the embodiment of the present invention;

[0029] Figure 2 It is the schematic diagram of the distance error calibration model established in the embodiment of the present invention;

[0030] Figure 3 It is the flow chart of the method for eliminating redundant parameters in the robot exponential product model. Detailed Embodiments

[0031] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the drawings. Taking the UR5 robot as an example, a robot exponential product model is established, the error calibration model adopts the distance error model, and the kinematic redundant parameters in the model are eliminated. The specific steps are as follows:

[0032] Step 1, establish a robot kinematic model;

[0033] According to the Figure 1 shown structure and dimension diagram of the UR5 robot, establish a robot exponential product model. Connect the base coordinate system {B} to the robot base, connect the tool coordinate system {T} to the end effector, and at the same time associate a twist with each joint. The joint twist of the i-th joint can be expressed as:

[0034]

[0035] where v i =-ω i ×r i , is the skew-symmetric matrix of ω i , and can be expressed as:

[0036]

[0037] r i =[r 1i r 2i r 3i T ​Denote the coordinates of the origin of each joint axis in the base coordinate system; ω i = [ω 1i ω 2i ω 3i T is the direction vector of each joint axis in the base coordinate system; is the screw coordinate of the joint screw.

[0038] For a revolute joint, the link transformation matrix from link i - 1 to link i can be described as:

[0039]

[0040] Therefore, the kinematic model of the robot is:

[0041]

[0042] where is the initial screw of the robot, calculated from the initial configuration of the robot, R(θ) ∈ R 3×3 and p(θ) ∈ R 3×1 are the attitude rotation matrix and the position translation matrix respectively.

[0043] The established robot POE model contains the following kinematic parameters:

[0044] θ1, θ2, …, θ6, ω1, ω2, …, ω6, r1, r2, …, r6, ξ st , where

[0045] ω i = [ω 1i ω 2i ω 3i T , r i = [r 1i r 2i r 3i T ,

[0046] ξ st = [ξ st1 ξ st2 ξ st3 ξ st4 ξ st5 ξ st6 . Therefore, there are 48 kinematic parameters in the model.

[0047] Step 2, establish the robot error calibration model;

[0048] Taking the total differential of the kinematic equation of the robot, the position error calibration model of the robot can be obtained:

[0049] ​​​Δp = HΔx#(15)

[0050] Where Δp is the end - position error, H is the Jacobian matrix of the position - error calibration model, and Δx is the kinematic - parameter error.

[0051] Next, establish the distance - error calibration model of the robot. As Figure 2 shown, assume that in the base coordinate system, the theoretical positions of the robot's end - effector reaching two points are P T,i (x T,i y T,i z T,i ) and P T,j (x T,j y T,j z T,j ), and the actual positions are P A,i (x A,i y A,i z A,i ) and P A,j (x A,j y A,j z A,j ). The difference between the two vectors of the theoretical distance and the actual distance can be expressed as:

[0052]

[0053] Therefore, there can be the following dot - product of vectors:

[0054]

[0055] Where θ is the angle between the vectors l T(i,j) and l A(i,j) . Since the robot error is a small value in practice, the two vectors of the theoretical distance and the actual distance are nearly parallel, and cosθ≈0. Therefore, the above formula can be approximated as:

[0056]

[0057] Where Δl i,j is the error between the theoretical distance and the actual distance. The above formula can be derived as:

[0058]

[0059] Combined with the position - error model of the robot, the distance - error calibration model is derived as:

[0060]

[0061] Where H j and H i are the Jacobian matrices under the corresponding position - error models of point i and point j respectively, It is the Jacobian matrix under the distance error calibration model.

[0062] Step 3: Calculate the Jacobian matrix J according to the robot joint variable measurement dataset and the robot error calibration model.

[0063]

[0064] where p is the number of data in the robot joint variable measurement dataset, and j a,b represents the sub-matrix of the Jacobian matrix calculated by differentiating the i-th kinematic parameter according to the a-th group of data. J i represents the i-th column of the Jacobian matrix, which corresponds to the i-th kinematic parameter of the robot.

[0065] The joint variable measurement dataset is recorded in the experiment. In the experiment, 500 groups of joint angles are measured at different poses of the robot in the working space of the UR5 robot as the measurement dataset.

[0066] Step 4: Find the columns in the Jacobian matrix J that are all zero, and update the Jacobian matrix after removing the corresponding ineffective parameters.

[0067] There are 9 columns in the Jacobian matrix J whose elements are all zero, and the corresponding ineffective parameters are θ1, θ6, ω 62 , r 13 , r 22 , r 32 , r 42 , r 53 , r 62 , After removing them, update the Jacobian matrix.

[0068] Step 5: Calculate the correlation coefficient between any two columns in the Jacobian matrix J.

[0069]

[0070] where r = ±1 indicates that there is a linear relationship between J i and J j , and the kinematic parameters corresponding to J i and J j are pairwise correlated parameters. For pairwise correlated parameters, only one of them is retained as an independent parameter, and the other parameters are removed.

[0071] According to the calculation results, a total of 4 pairwise correlated parameters are removed, which are ω 61 , ω 63 , ξ st3 , ξ st4 , and update the Jacobian matrix;

[0072] Step 6: Calculate one column J in the Jacobian matrix J column by column. iWith the other columns J1, J2, …, J i-1 , J i+1 , …, J n The multiple correlation coefficient R between them is calculated as follows:

[0073] First, use J i and J1, J2, …, J i-1 , J i+1 , …, J n to perform multiple linear regression, and obtain:

[0074]

[0075] It should be noted that when performing multiple correlation analysis on the Jacobian matrix, a constant term is not allowed. Therefore, the constant term is removed here for multiple linear regression:

[0076]

[0077] (2) Then calculate the simple correlation coefficient between J i and . This simple correlation coefficient is the multiple correlation coefficient between J i and J1, J2, …, J i-1 , J i+1 , …, J n . The calculation formula for the multiple correlation coefficient is:

[0078]

[0079] The value range of the multiple correlation coefficient R is [0, 1]. The larger R is, the higher the degree of correlation between J i and J1, J2, …, J i-1 , J i+1 , …, J n . When R = 1, it means that J i and J1, J2, …, J i-1 , J i+1 , …, J n are in a completely linear relationship. The kinematic parameter corresponding to J i is called a multiple correlation parameter, and the Jacobian matrix is updated after removing this parameter.

[0080] According to the calculation results, a total of 11 multiple correlation parameters are removed, which are θ2, θ3, θ4, θ5, ω 11 , ω, ω 12 , ω 21 , r 11 , r 12 , r 23 , ξ st1 .

[0081] Step 7: Since all multiple correlation coefficients \(R\neq1\) and all redundant parameters have been eliminated, the process ends.

[0082] According to the elimination results of the above steps, a total of 24 redundant parameters have been eliminated, as shown in Table 1.

[0083] The elimination results can be verified by calculating the rank of the Jacobian matrix: before elimination, there were a total of 48 parameters and the rank of the Jacobian matrix was 24; after elimination, there were a total of 24 parameters and the rank of the Jacobian matrix was 24. This proves the correctness of the elimination results.

[0084] Table 1 Redundant parameters in the exponential product model of the robot under distance error calibration

[0085]

[0086] The condition number is an index to measure the robustness of the robot kinematic parameter calibration system. The smaller the condition number, the better the robustness of the system. According to the calculation, by eliminating redundant parameters, the condition number of the Jacobian matrix has decreased from \(1.0436\times10\) 36 to \(9.1605\times10\) 4 , indicating that the robustness of the robot kinematic parameter calibration system has been significantly improved.

Claims

1. A method for eliminating redundant parameters in the exponential product model of a robot, which is used for eliminating redundant parameters in robot kinematic parameter calibration, is characterized in that, Specifically, it includes the following steps: Step 1: Establish a kinematic model of the robot; Step 2: Establish an error calibration model of the robot; Step 3: Calculate the Jacobian matrix J according to the robot joint variable measurement data set and the robot error calibration model; In the said Step 3, the Jacobian matrix J is: where n is the number of robot kinematic parameters, p is the number of data in the measured dataset of robot joint variables, and j a,b represents the sub-matrix of the Jacobian matrix calculated by differentiating the i-th kinematic parameter according to the a-th group of data, and J i represents the i-th column of the Jacobian matrix, which corresponds to the i-th kinematic parameter of the robot; Step 4: Find the columns in the Jacobian matrix J that are all zeros, and eliminate the corresponding ineffective parameters; Step 5: Calculate the correlation coefficient r between any two columns in the Jacobian matrix J, and eliminate the pairwise correlated parameters according to the calculation results; In the said Step 5, the calculation method of the correlation coefficient r between any two columns in the Jacobian matrix J is: Among them, J i and J j respectively represent the data of the i-th column and the j-th column of the Jacobian matrix J, and row(J) represents the number of rows of the Jacobian matrix J; When r = ±1, it represents J i There is a linear relationship with J j The kinematic parameters corresponding to J i and J j are pairwise-correlated parameters; for pairwise-correlated parameters, one of the parameters is retained as an independent parameter, and the remaining parameters are removed and the Jacobian matrix is updated; Step 6: Calculate the multiple correlation coefficient R between each column and other columns in the Jacobian matrix J, and eliminate the multiple correlated parameters according to the calculation results; In the said Step 6, the calculation method of the multiple correlation coefficient R between each column and other columns in the Jacobian matrix J is: Among them is the result of multiple linear regression of J i with other columns, represents the arithmetic mean of all data of J i ; When R = 1, it represents J i There is a completely linear relationship with other columns, and J is called i The corresponding kinematic parameter is a multi-correlated parameter. After removing this parameter, the Jacobian matrix is updated; Step 7: Judge whether all multiple correlation coefficients R≠1. If not, repeat Step 6. If so, end, indicating that all redundant parameters have been eliminated.

2. The method for eliminating redundant parameters in the robot exponential product model according to claim 1, characterized in that The said redundant parameters include ineffective parameters, pairwise correlated parameters or multiple correlated parameters, and are eliminated separately.

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