Industrial robot kinematics parameter identification method and system

Abstract

The invention provides an industrial robot kinematics parameter identification method and system, and relates to the technical field of industrial robots, and the method comprises the steps: obtaining theoretical kinematics parameters of an industrial robot, and joint coordinates and actual position coordinates of a plurality of measurement points; based on the theoretical kinematics parameters and the joint coordinates, a linear distance error equation set with kinematics parameter errors as unknown numbers is constructed; performing singular value decomposition on the coefficient matrix of the linear distance error equation set based on an L curve method and a triangle area criterion, and determining an optimal truncation point; according to the optimal truncation point, truncation singular value decomposition regularization processing is carried out on the coefficient matrix, and an equation set after regularization processing is obtained; according to the regularized equation set, the identification result of the kinematics parameter error is determined; and the identification result is ensured to be accurate and the actual identifiability of the parameters can be reflected.

Description

Technical Field

[0001] This application generally relates to the field of industrial robot technology, and specifically to a method and system for identifying kinematic parameters of an industrial robot. Background Technology

[0002] Industrial robots play a vital role in manufacturing and are increasingly widely used. Due to manufacturing tolerances, the actual kinematic parameters of an industrial robot after assembly may slightly deviate from the theoretically designed kinematic parameters. This affects the robot's absolute positioning accuracy. Wear and tear from long-term use also impacts kinematic parameters. Therefore, kinematic calibration of industrial robots is necessary to improve their accuracy.

[0003] Traditional methods for calibrating the kinematic parameters of industrial robots typically transform parameter identification into solving a system of linear equations. However, in practical applications, the coefficient matrix of these linear equations often exhibits ill-conditioned characteristics, making the solution process highly sensitive to measurement noise and affecting the accuracy and stability of parameter identification. To overcome this ill-conditioned problem, truncated singular value decomposition (SVD) is commonly used for regularization. However, the selection of the truncation threshold in SVD relies solely on human experience, lacking adaptability and making it difficult to achieve an optimal balance between preserving effective information and suppressing noise. Therefore, we propose a method and system for identifying kinematic parameters of industrial robots to address the aforementioned problems. Summary of the Invention

[0004] In view of the above-mentioned defects or deficiencies in the prior art, it is desirable to provide a method and system for identifying kinematic parameters of industrial robots that improves the reliability of identification.

[0005] In a first aspect, this application provides a method for identifying kinematic parameters of an industrial robot, comprising the following steps: The theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points are obtained; the theoretical kinematic parameters include at least the joint offset, link length, link offset and link torsion angle of the industrial robot. Based on the theoretical kinematic parameters and the joint coordinates, a system of linear distance error equations is constructed with the kinematic parameter error as the unknown. Based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equation system to determine the optimal cutoff point. Based on the optimal cutoff point, the coefficient matrix is ​​subjected to truncated singular value decomposition regularization to obtain the regularized system of equations. Based on the regularized equation set, the identification results of the kinematic parameter errors are determined; the identification results include the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

[0006] According to the technical solution provided in the embodiments of this application, based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equation system to determine the optimal cutoff point. Specifically, this includes the following steps: Based on the coefficient matrix of the linear distance error equation system, calculate the residual norm and solution norm corresponding to the multiple candidate cutoff points respectively; Plot the L-curve with the logarithm of the residual norm as the x-axis and the logarithm of the solution norm as the y-axis. Calculate the area of ​​the triangle formed by connecting each candidate cutoff point on the L-curve to the first and last endpoints of the L-curve, and take the candidate cutoff point with the largest triangle area as the optimal cutoff point.

[0007] According to the technical solution provided in the embodiments of this application, the method further includes the following steps: Calculate the parametric significance measure for each of the kinematic parameters, which is used to quantify the size of the projection of the kinematic parameter into the effective solution space retained after truncation; Based on the parameter significance measure, unidentifiable parameters in the linear distance error equation system are identified and removed.

[0008] According to the technical solution provided in the embodiments of this application, identifying and removing unidentifiable parameters in the linear distance error equation system specifically includes the following steps: Delete the column vector corresponding to the unidentifiable parameter from the coefficient matrix of the linear distance error equation system; and after determining the identification result of the kinematic parameter error, assign zero value to the position corresponding to the removed unidentifiable parameter.

[0009] According to the technical solution provided in the embodiments of this application, the significance measure of the parameter is determined according to the following steps: The parameter significance measure is obtained by performing a dot product operation between the column vectors of the right singular matrix obtained after the singular value decomposition and the unit basis vectors.

[0010] According to the technical solution provided in the embodiments of this application, a system of linear distance error equations with kinematic parameter errors as unknowns is constructed, specifically including the following steps: Based on the robot's kinematics model, a position error model is derived that transforms kinematic parameter errors into robot end-effector position errors. A distance error model is constructed based on the actual and theoretical distances between each pair of the multiple measurement points. Based on the position error model and the distance error model, the linear distance error equation system is obtained.

[0011] According to the technical solution provided in the embodiments of this application, the robot kinematic model is a DH model.

[0012] According to the technical solution provided in the embodiments of this application, the multiple measurement points are non-coplanarly distributed in the workspace of the industrial robot, and the number of measurement points is greater than the number of kinematic parameters to be identified.

[0013] According to the technical solution provided in the embodiments of this application, the method further includes the following steps: The error value corresponding to each kinematic parameter in the identification result is compensated into the theoretical kinematic parameter to obtain the calibrated kinematic parameter; the calibrated kinematic parameter is used to improve the absolute positioning accuracy of the industrial robot.

[0014] Secondly, this application provides an industrial robot kinematic parameter identification system for implementing the aforementioned industrial robot kinematic parameter identification method, the system comprising: The data acquisition module is used to acquire the theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points; the theoretical kinematic parameters include at least the joint offset, link length, link offset and link torsion angle of the industrial robot; The data processing module is used to construct a system of linear distance error equations with kinematic parameter errors as unknowns based on the theoretical kinematic parameters and the joint coordinates; perform singular value decomposition on the coefficient matrix of the linear distance error equations based on the L-curve method and the triangle area criterion to determine the optimal cutoff point; perform truncated singular value decomposition regularization on the coefficient matrix according to the optimal cutoff point to obtain a regularized system of equations; and determine the identification result of the kinematic parameter errors according to the regularized system of equations. The identification result includes the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

[0015] As can be seen from the above technical solution, this application has at least the following beneficial effects: This application provides a method for identifying kinematic parameters of an industrial robot, including acquiring the theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points; the theoretical kinematic parameters include at least the joint offset, link length, link offset, and link torsion angle of the industrial robot; based on the theoretical kinematic parameters and joint coordinates, a system of linear distance error equations is constructed with kinematic parameter errors as unknowns; based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equations to determine the optimal cutoff point; according to the optimal cutoff point, the coefficient matrix is ​​subjected to truncated singular value decomposition regularization processing to obtain the regularized equations; based on the regularized equations, the identification result of the kinematic parameter errors is determined; the identification result includes the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

[0016] This application provides fundamental data support for subsequent modeling by acquiring theoretical kinematic parameters and joint coordinates and actual position coordinates of multiple measurement points, ensuring that the data covers key robot motion parameters and actual operating states. Based on theoretical kinematic parameters and joint coordinates, kinematic parameter errors are treated as unknowns to construct a linear distance error equation system, transforming the parameter identification problem into a quantifiable and solvable linear equation system problem. Subsequently, to address the ill-conditioned problem of the coefficient matrix of the linear distance error equation system, the L-curve method and triangle area criterion are introduced to perform singular value decomposition on the coefficient matrix, determining the optimal cutoff point and achieving precise selection of the cutoff threshold, balancing information preservation and noise suppression. Then, based on the optimal cutoff point, the coefficient matrix is ​​subjected to truncated singular value decomposition regularization processing to remove noise interference corresponding to small singular values, resulting in a numerically stable regularized equation system. Finally, solving the regularized equation system not only determines the error value corresponding to each kinematic parameter, but also generates unidentifiable parameter identification information by analyzing the distribution of parameters in the singular value decomposition null space, thereby ensuring that the identification results are both accurate and reflect the actual identifiability of the parameters. Attached Figure Description

[0017] Other features, objects, and advantages of this application will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings.

[0018] Figure 1 A flowchart for a method to identify kinematic parameters of industrial robots.

[0019] Figure 2 A structural diagram of an industrial robot kinematic parameter identification system.

[0020] Figure 3 This is an example diagram of the L-curve.

[0021] Figure 4 This is an example diagram of a zero-space heatmap.

[0022] The diagram is labeled as follows: 10, data acquisition module; 20, data processing module. Detailed Implementation

[0023] The present application will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings.

[0024] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0025] like Figure 1 As shown, this application provides a method for identifying kinematic parameters of an industrial robot, including the following steps: S100. Obtain the theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points; the theoretical kinematic parameters include at least the joint offset, link length, link offset and link torsion angle of the industrial robot.

[0026] Theoretical kinematic parameters refer to the ideal parameters during the industrial robot design phase and serve as the benchmark for calculating parameter errors. Joint offset refers to the angular deviation of each joint relative to the reference coordinate system in its zero-position state, and is the benchmark value for calculating joint angles. Link length refers to the vertical distance between two adjacent joint axes, which determines the extension length of the link in space. Link offset refers to the distance from the origin of the current joint coordinate system to the origin of the next joint coordinate system along the direction of the current joint axis. Link torsion angle refers to the angle between the current joint axis and the next joint axis, which reflects the spatial attitude relationship between adjacent joint axes. Here, zero-position state refers to the initial state of the industrial robot. For example, the third joint being in a zero-position state means that the third joint is displayed as 0 degrees on the industrial robot teach pendant. The reference coordinate system refers to the joint coordinate system of each joint in the kinematic model; an industrial robot has as many joint coordinate systems as it has joints. For example, an industrial robot typically has 6 joints, so it has 6 joint coordinate systems.

[0027] The measurement points are obtained, for example, using a laser tracker, and the number of measurement points is, for example, 64. The selection of the spatial position of the measurement points affects the kinematic calibration. For example, one measurement point represents one measurement configuration of the industrial robot, and 64 measurement points represent 64 measurement configurations. The selection of these 64 positions was determined through extensive simulations to fully reflect the kinematic errors of the industrial robot.

[0028] Theoretical kinematic parameters are obtained, for example, from robot design drawings. Kinematic parameter calibration measures the error between the design and actual values. Industrial robot parts have tolerances during manufacturing and errors during assembly, so the final industrial robot requires kinematic calibration. For example, the actual link length from joint one to joint two may have a 0.1mm error compared to the theoretical link length on the design drawings. This error is the kinematic parameter error and needs to be calibrated and compensated for in the industrial robot's control system.

[0029] Multiple measurement points are distributed non-coplanarly within the workspace of the industrial robot, and the number of measurement points is greater than the number of kinematic parameters to be identified.

[0030] Joint coordinates refer to the angles of the six joints. Generally, industrial robots have six joints. Once the angles of these six joints are determined, the position of the robot's end effector is also determined. Therefore, the angles of these six joints can be called joint coordinates.

[0031] Actual position coordinates refer to the actual spatial position of the robot's end effector at the measurement point. They can be represented by the X, Y, and Z coordinates of the Cartesian coordinate system, such as X coordinate = 300mm, Y coordinate = 200mm, and Z coordinate = 500mm. Actual position coordinates can be acquired using a laser tracker or a binocular camera.

[0032] S200. Based on theoretical kinematic parameters and joint coordinates, a set of linear distance error equations is constructed with kinematic parameter errors as unknowns.

[0033] Constructing a system of linear distance error equations with kinematic parameter errors as unknowns includes the following steps: Based on the robot's kinematics model, a position error model is derived that transforms kinematic parameter errors into robot end-effector position errors. A distance error model is constructed based on the actual and theoretical distances between multiple measurement points; Based on the position error model and the distance error model, a set of linear distance error equations is obtained.

[0034] It should be noted that the robot's kinematic model, for example, is a Denavit-Hartenberg (DH) model, which is constructed using the DH parametric method. Based on the DH model, the transformation relationship between adjacent joint coordinate systems is described by joint angles, link offsets, link lengths, and link torsion angles. ;Formula (1) in, This is the homogeneous transformation matrix between adjacent joint coordinate systems. The number of joints. , , , , For the first Joint offset of each joint. For the first Linkage offset of each joint. For the first The length of the link in each joint For the first The torsion angle of each joint Rotation around the z-axis The rotation matrix, Translation along the z-axis The translation matrix, Translation along the x-axis The translation matrix, Rotation along the x-axis The rotation matrix.

[0035] Multiplying each homogeneous transformation matrix on the left in sequence yields the homogeneous transformation matrix from the robot's base coordinate system to the robot's end effector coordinate system: ;Formula (2) in, For robot kinematics model, This is the homogeneous transformation matrix between adjacent joint coordinate systems.

[0036] The position error model for industrial robots includes models for the transformation from kinematic parameter errors to joint coordinate system errors, and models for the transformation from joint coordinate system errors to robot end-effector pose errors. In a given joint coordinate system, the pose error of the corresponding link end can be represented in two ways: one is by the translational errors along the three coordinate axes and the rotational errors around the three coordinate axes; the other is by the errors of the relevant kinematic parameters of that joint. By establishing the mathematical equivalence between these two representations, the transformation matrix from kinematic parameter errors to joint coordinate system pose errors can be derived.

[0037] In the In a joint coordinate system, set , , Let x, y, and z represent the small translation errors of the ends of the corresponding links along the x, y, and z axes, respectively. Then, the differential translation matrix can be expressed as: ;Formula (3) in, It is a differential translation matrix (a translation matrix along the x-axis, y-axis, and z-axis), which is used to construct a mathematical model of the position error of an industrial robot.

[0038] set up , , These represent the minute rotational errors of the corresponding link ends around the x, y, and z axes, respectively. Because... It is a tiny quantity; taking the limit, we get: ;Formula (4) The above , Similarly, this will not be elaborated further here. Ignoring second-order terms, the differential rotation matrix can be expressed as: ;Formula (5) in, This is a differential rotation matrix, used to construct a mathematical model of the position error of an industrial robot. Rotation about the x-axis The rotation matrix, Rotation about the y-axis The rotation matrix, Rotation about the z-axis The rotation matrix.

[0039] set up In the first In a joint coordinate system, the difference between the pose after differential motion at the joint end and the initial pose is given by the following formula: The joint pose error, expressed in terms of translation and rotation errors in a fixed coordinate system, is derived from the following formula. : ;Formula (6) ;Formula (7) Substituting formulas (3) and (5) into formula (7), we get: ;Formula (8) in, Let be a differential operator with reference to a fixed coordinate system, its representation is from arrive mathematical relationship, This is the homogeneous transformation matrix between adjacent joint coordinate systems.

[0040] If the differential error is represented by the coordinate system of motion, then the calculation formula is: ;Formula (9) in, It is a differential operator with reference to the moving coordinate system.

[0041] In the In a joint coordinate system, set , , , These represent the errors in joint angle, link offset, link length, and link torsion angle, respectively. Taking the partial derivatives of the kinematic model of the industrial robot using multivariate functions, then... The expression is: ;Formula (10) Substituting formula (1) into formula (10), we get: ;Formula (11) Based on the mathematical equivalence between formula (8) and formula (11), we can derive: ;Formula (12) Rearrange formula (12) into matrix form: ;Formula (13) in, This represents the positional error of the link end in the joint coordinate system. For kinematic parameter error, This is a matrix representing the kinematic parameter error to the joint coordinate system pose error, with reference to the motion coordinate system.

[0042] Formula (13) is the mathematical model for the transition from kinematic parameter error to joint coordinate system error. Then, by deriving the mathematical model for the transition from joint coordinate system error to robot end-effector pose error, the position error model from kinematic parameter error to robot end-effector position error can be obtained. Specifically, the following is set... The error matrix of the end-effector coordinate system relative to the robot base coordinate system is expressed as follows: ;Formula (14) in, For industrial robots Kinematic model of each joint In the first In the joint coordinate system of the nth joint, the nth joint Error matrices corresponding to each joint.

[0043] Expanding equation (14) and ignoring the second-order differential error, we get: ;Formula (15) The detailed process from formula (14) to formula (15) is as follows:

[0044] Unpack all the parentheses on the right side of the equals sign, multiply all the terms, and you get: ; Second-order differential errors are omitted, meaning that all terms on the right-hand side of the equation with two or more terms having dA are treated as 0 and removed.

[0045] ; and Similarly, take the right side of the equals sign... Written ,get: ; Both sides After eliminating it, you get: ; Thus, we obtain formula (15), which can be expressed using the summation symbol:

[0046] Substituting formula (8) into formula (15), we get: ;Formula (16) From formula (9), we can derive: ;Formula (17) in, This is the differential operator for the robot's end-effector pose error, with reference to the motion coordinate system.

[0047] Substituting formula (17) into formula (16), we get: ;Formula (18) in, The quantity generated during the calculation process means: with the moving coordinate system as a reference, the first... The differential operator for the robot end-effector pose error caused by the kinematic parameter error of each joint.

[0048] Set vector , , , , , They are respectively: ;Formula (19) Using vectors , , , , , express , , ,get: ;Formula (20) ;Formula (21) ;Formula (22) in, For the first Differential operators for each joint, The quantities generated during the calculation process are represented with reference to the moving coordinate system. Differential operator for robot end-effector pose error caused by kinematic parameter errors of individual joints For the first A kinematic model of the joints to the robot's end effector.

[0049] Based on the derivation from formula (18) to formula (22), we get: ;Formula (23) Substituting formulas (20), (21), and (22) into formula (23), we get: ;Formula (24) Based on the mathematical equivalence between formulas (21) and (24), we obtain: ;Formula (25) The matrix form of formula (25) is as follows: ;Formula (26) in, The quantities generated during the calculation process are represented with reference to the moving coordinate system. The robot end-effector pose error caused by the kinematic parameter errors of each joint This represents the positional error of the link end in the joint coordinate system. From arrive The matrix is ​​the mathematical model from the joint coordinate system link end pose error to the robot end pose error.

[0050] Substituting formula (13) into formula (26), we get: ;Formula (27) Based on formulas (18) and (27), we can derive the following: ;Formula (28) in, The robot end-effector pose error is caused by the kinematic parameter errors of all joints with reference to a fixed coordinate system. In actual measurement, the observed position coordinates include not only the error of translation along the coordinate axes, but also the error of rotation around the coordinate axes. Therefore, based on the derivation of formulas (17) and (28), we obtain: ;Formula (29) in, With reference to a fixed coordinate system, the first The robot end-effector position error at each measurement point for The top-left 3x3 submatrix for The first three lines, For the first The matrix of measurement points from kinematic parameter errors to robot end-effector position errors represents the mathematical relationship between kinematic parameter errors and industrial robot end-effector position errors, i.e., the industrial robot position error model.

[0051] Then, an industrial robot distance error model is established. Specifically, the relationship between the distance error between two measurement points and the position error between the two measurement points is as follows: ;Formula (30) Substituting formula (29) into formula (30), we get: ;Formula (31) in, To the kinematic parameter error to the first The measurement point and the first A matrix of distance errors at each measurement point.

[0052] All Vertical arrangement yields a matrix , This is a matrix from kinematic parameter errors to all distance errors, i.e., the distance error model of an industrial robot.

[0053] The above matrix This refers to the coefficient matrix of the linear distance error equation system. For example, the coefficient matrix has 24 columns, corresponding to 24 kinematic parameters. If there are 64 measurement points, then combining each of the 64 measurement points in pairs can form 2016 distance errors, and the coefficient matrix has 2016 rows.

[0054] S300. Based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equation system to determine the optimal cutoff point.

[0055] Among them, based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equation system to determine the optimal cutoff point. The specific steps include: Based on the coefficient matrix of the linear distance error equation system, calculate the residual norm and solution norm corresponding to multiple candidate cutoff points respectively; Plot the L-curve with the logarithm of the residual norm as the x-axis and the logarithm of the solution norm as the y-axis. Calculate the area of ​​the triangle formed by connecting each candidate cutoff point on the L-curve to the first and last endpoints of the L-curve, and take the candidate cutoff point with the largest triangle area as the optimal cutoff point.

[0056] It should be noted that Singular Value Decomposition (SVD) is a powerful and widely used matrix factorization technique in linear algebra. For any m×n real matrix A, its SVD is defined as... ,in, A is an m×m orthogonal matrix, and its column vectors are called left singular vectors. These vectors form an orthonormal basis for the column space (Range) of A. It is an m×n rectangular diagonal matrix. The elements on its diagonal are called singular values, which are non-negative real numbers and are usually arranged in descending order. The off-diagonal elements of the matrix are all zero. yes The transpose, and A is an n×n orthogonal matrix whose column vectors are called right singular vectors; these vectors form a set of orthonormal bases for the row space of A.

[0057] Any matrix multiplication can be viewed as a combination of rotation and scaling transformations. Specifically, the transformation Ax of matrix A onto a vector x mainly includes the following steps: left multiplication. First, rotate vector x to the position of vector x. The matrix is ​​then built on an orthogonal basis spanned by the column vectors. The rotated vector is scaled to different degrees along each coordinate axis. The scaling ratio corresponds to the singular value. The larger the singular value, the stronger the influence or energy of matrix A in that direction; finally, left multiply... The scaled vector is then rotated back to the final target space. From this perspective, The essence is to find two sets of orthonormal bases ( and This means that the function of matrix A is simply to scale the coordinate axes between these two bases.

[0058] The core idea of ​​Truncated Singular Value Decomposition (TSVD) is low-rank approximation. In practical applications, singular values... The value will decrease rapidly, and many smaller singular values ​​may be close to zero. These smaller singular values ​​usually correspond to noise, details, or rounding errors in the data, rather than the main structure of the data. TSVD approximates the original matrix with a low-rank matrix by discarding the smaller singular values ​​and their corresponding left and right singular vectors.

[0059] Suppose that the rank of matrix A is Its corresponding SVD is the sum of all singular values: ;Formula (32) in, for The List, for The List.

[0060] Select a cutoff point ( < TSVD approximation matrix Defined as retaining only the previous one The largest singular values ​​and their corresponding singular vectors: ;Formula (33) Solving ill-conditioned linear systems In some cases, direct inversion may result in numerical instability. TSVD can be used to calculate a stable pseudo-inverter, ignoring singular values ​​close to zero to avoid noise amplification. The key to truncation lies in choosing the truncation threshold; truncating too much will result in excessive information loss, while truncating too little will contaminate the solution with noise.

[0061] Based on the singular value decomposition results, for each candidate cutoff point k, calculate the corresponding residual norm and solution norm: ;Formula (34) in, The residual norm reflects the fitting accuracy of the truncated system of equations, i.e., the degree of matching between the solution and the actual measured data. A smaller value indicates more useful information is retained and a better fit; a larger value indicates excessive truncation and loss of key information. The calculation logic of the residual norm is based on the geometric meaning of SVD, which is that after removing small singular values, the residuals mainly originate from the noise components corresponding to the truncated singular values.

[0062] ;Formula (35) in, The solution norm reflects the stability of the truncated parameter error solution. A smaller value indicates less noise interference and better stability; a larger value indicates more noise and poorer stability. The solution norm itself does not represent the noise level. However, a sudden change in the solution's 2-norm can indicate that the main components of the solution have shifted from useful information to disordered noise.

[0063] Plot the L-curve with the logarithm of the residual norm on the x-axis and the logarithm of the solution norm on the y-axis. The parametric expression for the L-curve is: ;Formula (36) Where k is a parameter, representing the 1st point, the 2nd point, ..., the kth point. Represents the x-axis. Let x be the x-coordinate of the k-th point. Represents the vertical axis. Let be the y-coordinate of the k-th point. Let be the norm of the residual, defined in equation (34). The norm of the solution is defined in equation (35).

[0064] The right side of the L-curve is the over-regularized region, where the residual norm is large, indicating that too much information has been truncated. The left side of the L-curve is the under-regularized region, where the solution norm increases sharply, indicating that noise dominates the solution. The inflection point is the optimal balance between information preservation and noise suppression.

[0065] like Figure 3 As shown, the inflection point detection of L-curves generally uses the curvature maximization criterion. However, the curvature maximization criterion requires numerical differentiation and is sensitive to noise. It is easy to misjudge local inflection points in over-regularized smooth segments as global inflection points. Therefore, the triangle criterion is used to find inflection points.

[0066] First, determine two fixed reference points on the L-curve, which will serve as the endpoints of the base of the triangle for calculating its area. As the first endpoint, This is the endpoint.

[0067] Define the curve point set as: ;Formula (37) Let the baseline vector of the triangle criterion be: ;Formula (38) point to the straight line vertical distance and , The cross product of vectors is proportional: ;Formula (39) Therefore, the area function can be defined as: ;Formula (40) Therefore, the optimal parameters are: ;Formula (41) The area obtained by formula (40) is twice the area of ​​the triangle, but this does not affect the solution of the optimal parameters. The geometric interpretation of formula (41) is to make Find the largest area. Iterate through all intermediate candidate cutoff points. Find the cutoff point corresponding to the largest area of ​​the triangle, which is the optimal cutoff point.

[0068] S400. Based on the optimal cutoff point, the coefficient matrix is ​​subjected to truncated singular value decomposition regularization to obtain the regularized system of equations.

[0069] After singular value decomposition, The diagonal elements (singular values) of the matrix are sorted in descending order. Larger singular values ​​correspond to the main structure of the data (useful information), while smaller singular values ​​correspond to measurement noise. Based on the optimal cutoff point, The singular values ​​in the matrix starting from the optimal cutoff point and those ordered thereafter are set to 0, resulting in the truncated singular value matrix; while retaining The optimal cutoff point of the matrix is ​​in the first column. The optimal truncation point of the matrix is ​​first identified. Using the truncated matrix, the regularized pseudoinverse of the coefficient matrix is ​​calculated, ultimately transforming the original system of equations into a regularized system. The solution of this regularized system is less affected by noise, and its numerical stability is significantly improved.

[0070] S500. Based on the regularized equation set, determine the identification results of the kinematic parameter errors; the identification results include the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

[0071] Furthermore, this method also includes the following steps: The error values ​​corresponding to each kinematic parameter in the identification results are compensated into the theoretical kinematic parameters to obtain the calibrated kinematic parameters; the calibrated kinematic parameters are used to improve the absolute positioning accuracy of industrial robots.

[0072] In particular, the regularized equation set obtained by solving step S400 directly yields the specific values ​​of the errors of each kinematic parameter. The theoretical parameters plus the parameter errors are used as the actual kinematic parameters of the industrial robot, providing a direct basis for subsequent robot calibration (parameter correction).

[0073] Based on the right singular matrix obtained from the singular value decomposition in step S300 Defined as a parameter significance measure, if the measure value of a parameter is 1, it means that the effect of the parameter on the end position error exists entirely in the null space formed after truncation. That is, changes in the parameter will not cause significant changes in the measured value (distance error), and therefore it is unidentifiable. In the final identification results, such parameters are marked as unidentifiable to avoid correcting invalid parameters in subsequent calibration, thereby improving calibration efficiency and accuracy.

[0074] Furthermore, this method also includes the following steps: Calculate the parametric significance measure for each kinematic parameter. The parametric significance measure is used to quantify the size of the projection of the kinematic parameter into the effective solution space after truncation. Based on the parameter significance measure, unidentifiable parameters in the linear distance error equation system are identified and removed.

[0075] It should be noted that the significance measure of the parameters is determined according to the following steps: The saliency measure of parameters is obtained by performing a dot product operation between the column vectors of the right singular matrix obtained after singular value decomposition and the unit basis vectors.

[0076] Specifically, the significance measure of the parameters is calculated according to the following formula: ;Formula (42) in, For the first Unit basis vectors for Matrix number The first column vector of the column vector Each component. It is after singular value truncation matrix.

[0077] like Figure 4 As shown, the numerical null space refers to the space after being truncated by the L-curve, from... arrive Small singularities, from arrive Since the numerical calculation of zero singular values ​​is non-zero, they are all forced to be set to zero. Become a matrix The actual rank, therefore, is assumed. Tolerance threshold According to the regularization formula, the solution... It can be viewed as a matrix A linear combination of column vectors, and In the matrix A null space is formed on the right side. That is, the Kth column is used as the dividing line. All the numbers on the right side are 0.

[0078] If the parameter significance measure An equal value of 1 indicates that the first... The first parameter exists entirely in the null space, so its influence on the measured value is extremely small. The parameters are unidentifiable.

[0079] For example, The proof that the i-th parameter is unrecognizable when the value equals 1 is as follows: Apply a perturbation to the null space and observe the effect of the null space on the system output when it is not truncated.

[0080] set up , ,but It can be represented as: ;Formula (43) Where H is the coefficient matrix of the system of equations to be solved.

[0081] Perform truncated singular value decomposition on H to obtain three matrices. Substituting into formula (43), we get: ;Formula (44) make , It is a column vector. Therefore: ;Formula (45) in, This represents the i-th singular value. express The i-th term, yes The transpose of the i-th column vector, with T in the upper right corner representing the transpose.

[0082] because It is an orthogonal matrix, therefore: ;Formula (46) Substituting formula (46) into formula (45) yields: ;Formula (47) in, It is the truncation threshold for truncating singular value decomposition.

[0083] Taking the square root, we get: ;Formula (48) As can be seen from formula (48), the influence of disturbances in the numerical null space direction on the system output is... Controls are negligible in ill-conditioned systems. This supports the truncation of the null space. However, if a parameter to be identified consists entirely of the numerical null space, then changes in this parameter are unlikely to have a significant impact on the measured value. This is the meaning of parameter significance measurement. Definition: ;Formula (49) in, It is defined. It is the first Unit basis vectors for Matrix number The first column vector of the column vector Each component.

[0084] The matrix is ​​an orthogonal matrix, and the magnitudes of its row vectors and column vectors are both 1 / 2. , combined As can be seen from the definition: when At that time, the first Each parameter to be identified is entirely composed of a null space.

[0085] set up ,satisfy , It is after singular value truncation matrix.

[0086] Because we need to observe Therefore, Changes should be concentrated on the components. ;because Since it exists only in the null space, perturbations to the solution only occur in the null space. Construct the perturbation vector: ;Formula (50) in, It is the first Unit basis vectors. It is after singular value truncation matrix.

[0087] For the perturbation solution The residuals are, It is the exact value of the solution, which only exists in theory. c is an arbitrary constant, and d is the column vector calculated by formula (50).

[0088] ;Formula (51) in, For noise, It is an arbitrary constant.

[0089] Substituting formula (50) into formula (51) yields: ;Formula (52) For parameters The changes in the perturbation solution are as follows: ;Formula (53) From formulas (52) and (53), it can be seen that when the solution is entirely composed of the null space, and only the direction of the null space changes, the increment of the residual is... ,exist When extremely small, it can be ignored. And the parameters... The offset is , Linear growth. Therefore... It cannot be uniquely determined.

[0090] After removing redundant columns, the condition number of the coefficient matrix is ​​reduced (the ill-conditioning is alleviated), and the accuracy and stability of the regularization solution are significantly improved, ensuring that the error results of the identifiable parameters are more reliable.

[0091] In addition, identifying and removing unidentifiable parameters in the linear distance error equation system involves the following steps: In the coefficient matrix of the linear distance error equation system, delete the column vectors corresponding to the unidentifiable parameters; and after determining the identification results of the kinematic parameter errors, assign zero values ​​to the positions corresponding to the removed unidentifiable parameters.

[0092] Theoretically, this approach eliminates redundant interference from unidentifiable parameters on the equation system, ensuring stable and accurate solutions for identifiable parameters and resolving the issue of low accuracy in ill-conditioned matrix solutions. From an engineering perspective, it restores the complete dimensions of the parameter error vector, meeting the data format requirements for robot calibration and avoiding application obstacles caused by dimensional mismatches in the solution results. This combination ensures both the mathematical rigor of the identification process and the engineering practicality of the output results, laying a crucial foundation for improving the absolute positioning accuracy of industrial robots through parameter correction.

[0093] This application provides fundamental data support for subsequent modeling by acquiring theoretical kinematic parameters and joint coordinates and actual position coordinates of multiple measurement points, ensuring that the data covers key robot motion parameters and actual operating states. Based on theoretical kinematic parameters and joint coordinates, kinematic parameter errors are treated as unknowns to construct a linear distance error equation system, transforming the parameter identification problem into a quantifiable and solvable linear equation system problem. Subsequently, to address the ill-conditioned problem of the coefficient matrix of the linear distance error equation system, the L-curve method and triangle area criterion are introduced to perform singular value decomposition on the coefficient matrix, determining the optimal cutoff point and achieving precise selection of the cutoff threshold, balancing information preservation and noise suppression. Then, based on the optimal cutoff point, the coefficient matrix is ​​subjected to truncated singular value decomposition regularization processing to remove noise interference corresponding to small singular values, resulting in a numerically stable regularized equation system. Finally, solving the regularized equation system not only determines the error value corresponding to each kinematic parameter, but also generates unidentifiable parameter identification information by analyzing the distribution of parameters in the singular value decomposition null space, thereby ensuring that the identification results are both accurate and reflect the actual identifiability of the parameters.

[0094] like Figure 2 As shown, this application provides an industrial robot kinematic parameter identification system for implementing the above-described industrial robot kinematic parameter identification method. The system includes: The data acquisition module 10 is used to acquire the theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points; the theoretical kinematic parameters include at least the joint offset, link length, link offset and link torsion angle of the industrial robot. The data processing module 20 is used to construct a system of linear distance error equations with kinematic parameter errors as unknowns based on theoretical kinematic parameters and joint coordinates; to perform singular value decomposition on the coefficient matrix of the linear distance error equations based on the L-curve method and the triangle area criterion to determine the optimal cutoff point; to perform truncated singular value decomposition regularization on the coefficient matrix according to the optimal cutoff point to obtain the regularized equations; and to determine the identification results of the kinematic parameter errors based on the regularized equations. The identification results include the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

[0095] The above description is merely a preferred embodiment of this application and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in this application is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the inventive concept. For example, technical solutions formed by substituting the above features with (but not limited to) technical features with similar functions disclosed in this application.

Claims

1. A method for identifying kinematic parameters of an industrial robot, characterized in that, Includes the following steps: The theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points are obtained; the theoretical kinematic parameters include at least the joint offset, link length, link offset and link torsion angle of the industrial robot. Based on the theoretical kinematic parameters and the joint coordinates, a system of linear distance error equations is constructed with the kinematic parameter error as the unknown. Based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equation system to determine the optimal cutoff point. Based on the optimal cutoff point, the coefficient matrix is ​​subjected to truncated singular value decomposition regularization to obtain the regularized system of equations. Based on the regularized equations, the identification results of the kinematic parameter errors are determined; The identification results include the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

2. The method for identifying kinematic parameters of an industrial robot according to claim 1, characterized in that, Based on the L-curve method and the triangle area criterion, singular value decomposition is performed on the coefficient matrix of the linear distance error equation system to determine the optimal cutoff point. This includes the following steps: Based on the coefficient matrix of the linear distance error equation system, calculate the residual norm and solution norm corresponding to the multiple candidate cutoff points respectively; Plot the L-curve with the logarithm of the residual norm as the x-axis and the logarithm of the solution norm as the y-axis. Calculate the area of ​​the triangle formed by connecting each candidate cutoff point on the L-curve to the first and last endpoints of the L-curve, and take the candidate cutoff point with the largest triangle area as the optimal cutoff point.

3. The method for identifying kinematic parameters of an industrial robot according to claim 1, characterized in that, The method further includes the following steps: Calculate the parametric significance measure for each of the kinematic parameters, which is used to quantify the size of the projection of the kinematic parameter into the effective solution space retained after truncation; Based on the parameter significance measure, unidentifiable parameters in the linear distance error equation system are identified and removed.

4. The method for identifying kinematic parameters of an industrial robot according to claim 3, characterized in that, Identifying and removing unidentifiable parameters from the linear distance error equation system specifically includes the following steps: Delete the column vector corresponding to the unidentifiable parameter from the coefficient matrix of the linear distance error equation system; and after determining the identification result of the kinematic parameter error, assign zero value to the position corresponding to the removed unidentifiable parameter.

5. A method for identifying kinematic parameters of an industrial robot according to claim 3 or 4, characterized in that, The significance measure of the parameter is determined according to the following steps: The parameter significance measure is obtained by performing a dot product operation between the column vectors of the right singular matrix obtained after the singular value decomposition and the unit basis vectors.

6. The method for identifying kinematic parameters of an industrial robot according to claim 1, characterized in that, Constructing a system of linear distance error equations with kinematic parameter errors as unknowns includes the following steps: Based on the robot's kinematics model, a position error model is derived that transforms kinematic parameter errors into robot end-effector position errors. A distance error model is constructed based on the actual and theoretical distances between each pair of the multiple measurement points. Based on the position error model and the distance error model, the linear distance error equation system is obtained.

7. The method for identifying kinematic parameters of an industrial robot according to claim 6, characterized in that, The robot's kinematic model is the DH model.

8. The method for identifying kinematic parameters of an industrial robot according to claim 1, characterized in that, The multiple measurement points are non-coplanarly distributed within the workspace of the industrial robot, and the number of measurement points is greater than the number of kinematic parameters to be identified.

9. The method for identifying kinematic parameters of an industrial robot according to claim 1, characterized in that, The method also includes the following steps: The error value corresponding to each kinematic parameter in the identification result is compensated into the theoretical kinematic parameter to obtain the calibrated kinematic parameter; the calibrated kinematic parameter is used to improve the absolute positioning accuracy of the industrial robot.

10. An industrial robot kinematic parameter identification system, used to implement the industrial robot kinematic parameter identification method according to any one of claims 1-9, characterized in that, The system includes: The data acquisition module is used to acquire the theoretical kinematic parameters of the industrial robot and the joint coordinates and actual position coordinates of multiple measurement points; the theoretical kinematic parameters include at least the joint offset, link length, link offset and link torsion angle of the industrial robot; The data processing module is used to construct a system of linear distance error equations with kinematic parameter errors as unknowns based on the theoretical kinematic parameters and the joint coordinates; perform singular value decomposition on the coefficient matrix of the linear distance error equations based on the L-curve method and the triangle area criterion to determine the optimal cutoff point; perform truncated singular value decomposition regularization on the coefficient matrix according to the optimal cutoff point to obtain a regularized system of equations; and determine the identification result of the kinematic parameter errors according to the regularized system of equations. The identification result includes the error value corresponding to each kinematic parameter and the identification information of unidentifiable parameters.

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