SCARA robot arm length and zero point calibration method based on support vector regression

Abstract

This invention discloses a SCARA robot arm length and zero point calibration method using support vector regression. By establishing a kinematic model for the SCARA robot, recording joint angle data at preset points on a calibration plate, and constructing an overdetermined homogeneous linear equation system, the offset between the arm length and the zero point is initially solved. To address nonlinear errors that are difficult to handle with traditional calibration methods, this invention introduces an SVR model and utilizes radial basis kernel functions (RBFs) for error modeling and compensation, enabling dynamic error correction for the robot under different environments. This method significantly improves calibration accuracy by collecting actual error data within the robot's workspace in real time, reduces human intervention, and ensures the robot's precision stability during high-speed operation and long-term operation. Compared to existing technologies, this invention effectively handles nonlinear errors in complex environments through automated calibration and error compensation, improving industrial production efficiency and reducing downtime. It exhibits strong robustness and broad application prospects.

Description

Technical Field

[0001] The present invention relates to the field of robotics technology, and in particular to a SCARA robot arm length and zero point calibration method based on support vector regression. Background Art

[0002] The SCARA (Selective Compliance Assembly Robot Arm) is an industrial robot designed for fast and precise horizontal operations. It is widely used in industries such as electronic product assembly, material handling, precision packaging, and dispensing. The SCARA robot's design features a highly rigid vertical axis and a highly flexible horizontal axis. It can move flexibly within a two-dimensional plane while maintaining vertical rigidity, making it suitable for industrial operations requiring high precision and speed. The accuracy and repeatability of SCARA robots' operations directly depend on the calibration of their arm length and zero position. However, due to minor errors in the manufacturing and assembly processes, the robot is prone to arm length deviations and zero position offsets during operation, resulting in inconsistencies between its actual working path and the theoretical trajectory, seriously affecting operational accuracy.

[0003] In order to ensure the operating accuracy of SCARA robots, the industry generally uses several calibration methods to calibrate the arm length and zero point position. The more common methods in the existing technology include:

[0004] 1. Laser rangefinder calibration: The robot arm length and zero point are calibrated using a laser rangefinder. This method offers the advantages of high precision and non-contact measurement, but its disadvantages are that it requires high-precision measurement equipment, which is expensive. Furthermore, in complex industrial environments, it is susceptible to interference from the laser path, resulting in unstable measurement results.

[0005] 2. Mechanical alignment calibration method: Manually or semi-automatically adjust the arm length and zero point of the SCARA robot. This method is simple and easy to operate, but due to its reliance on manual operation, it has low accuracy. The calibration process is time-consuming and susceptible to human factors, resulting in poor consistency and repeatability.

[0006] 3. Based on mathematical modeling and simulation: The deviation between arm length and zero position is inferred from the known motion trajectory and the desired position, and the model is used to correct the error. However, this method relies on the accuracy of the model and the performance of the simulation software. If the simulation model contains errors, the actual results often fail to meet the high precision requirements. In addition, this method requires high computing power and the calibration process is complex.

[0007] In addition, the disadvantages of existing technologies are that the laser ranging method is costly and sensitive to the environment, and is easily affected by external interference and the measurement results; the mechanical alignment method has low accuracy and efficiency, and is easily affected by human operation errors, and cannot meet high-precision industrial needs; although mathematical modeling and simulation methods can partially solve the errors, they are limited by the accuracy of the model and simulation conditions, and it is difficult to handle complex nonlinear errors, and the computational burden is heavy.

[0008] To address these issues, the industry is increasingly adopting calibration methods based on error compensation. These methods compensate for deviations in the robot's arm length and zero-point position through data acquisition and modeling. However, existing error compensation methods are typically based on linear models and cannot effectively address the nonlinear errors caused by factors such as joint wear and assembly errors during actual robot operation. Furthermore, existing compensation methods typically only work under static conditions after calibration and lack the ability to adjust dynamically, resulting in an inability to maintain stable accuracy during high-speed operation.

[0009] Therefore, how to improve the accuracy of SCARA robot arm length and zero point calibration and handle dynamic and nonlinear error compensation has become the technical problem to be solved by the present invention. Summary of the Invention

[0010] The technical problem solved by the present invention is to provide a SCARA robot arm length and zero point calibration method based on support vector regression in response to the defects existing in the above-mentioned prior art, so as to solve the problems of low calibration accuracy, insufficient error compensation, and inability to handle nonlinear and dynamic errors proposed in the above-mentioned background technology.

[0011] In order to solve the above technical problems, the technical solutions adopted by the present invention are as follows:

[0012] A SCARA robot arm length and zero point calibration method based on support vector regression, the method comprising the following steps:

[0013] Step 1: Establish the kinematic equations of the SCARA robot: Based on the structure of the upper and lower arms of the SCARA robot, define its kinematic model, where:

[0014] The length of the upper arm is L1, and the length of the lower arm is L2;

[0015] The first joint angle is θ1, and the second joint angle is θ2;

[0016] The position of the SCARA robot end in the workspace is determined by the following equation:

[0017]

[0018] Step 2: Calibration point selection and data collection: Align the calibration pin at the end of the SCARA robot with the preset points M and N on the calibration plate, record the joint angles θ1 and θ2 at different positions, and establish an error model for arm length and zero offset based on the measured angles;

[0019] Step 3: Preliminary solution based on overdetermined homogeneous linear equations: Based on the collected multiple calibration point positions and the corresponding joint angles, construct an overdetermined homogeneous linear equation system:

[0020] Ax=0

[0021] Among them, the matrix A is composed of the coordinates and angle information of each calibration point. By solving the eigenvector corresponding to the minimum eigenvalue of the equation group, the arm length error and zero offset value of the SCARA robot are preliminarily estimated;

[0022] Step 4: Construction and training of support vector regression (SVR) model:

[0023] Use multiple sets of data points randomly collected in the SCARA robot workspace to record the error between the actual position and the ideal position. The error is the residual error after calibration.

[0024] The error data set is divided into a training set and a test set, and regression analysis is performed using the support vector regression SVR model. The model uses the radial basis kernel function RBF, and its loss function is defined as:

[0025]

[0026] in, is the weight vector, is the regularization parameter, and is a slack variable, representing the prediction error;

[0027] Step 5, error compensation: Based on the training results of the SVR model, the position error of the SCARA robot is predicted in real time when the robot is moving, and the target position of the SCARA robot is compensated according to the predicted error value. The compensation formula is:

[0028]

[0029] in, is the error value predicted by the SVR model;

[0030] Step 6, calculation and adjustment: Based on the compensated SCARA robot end position, further adjust the arm length and zero point position to ensure high-precision operation of the robot in different working environments.

[0031] As a further solution of the present invention, the calibration points include at least two preset points in the workspace of the SCARA robot, and the calibration process is performed by aligning the calibration needle with the preset points in sequence and recording the joint angles.

[0032] As a further solution of the present invention, the overdetermined homogeneous linear equations are solved by the Jacobi method to obtain the eigenvector corresponding to the minimum eigenvalue, which is used to preliminarily calculate the arm length error and zero point offset.

[0033] As a further solution of the present invention, the support vector regression model uses a radial basis kernel function (RBF) to process the nonlinear error of the SCARA robot in the workspace.

[0034] As a further solution of the present invention, the data in the error data set is obtained by random sampling to ensure the diversity and representativeness of the data.

[0035] As a further solution of the present invention, the SVR model is trained by a sequential minimum optimization (SMO) algorithm to minimize the error and determine the optimal support vector.

[0036] Compared with the prior art, the present invention has the following beneficial effects:

[0037] 1. Accurate arm length and zero point calibration: This invention significantly improves the calibration accuracy of SCARA robot arm length and zero point by introducing support vector regression (SVR) and utilizing radial basis kernel (RBF) functions to address nonlinear errors. Compared to traditional laser ranging or mechanical alignment calibration methods, this invention automatically adapts to complex nonlinear error distributions, maintaining high accuracy even in the face of long-term robot operation, wear, or accumulated assembly errors. Through precise error modeling and regression, this invention not only reduces human interference but also reduces reliance on high-precision hardware equipment, lowering overall costs.

[0038] 2. Dynamic Error Compensation: In real-world industrial applications, SCARA robots often operate in high-speed and dynamic environments. Traditional calibration methods struggle to adapt to the dynamic errors introduced by these changing environments. This invention utilizes the SVR model, combined with real-time data acquisition, to perform dynamic error compensation during robot operation. This real-time compensation mechanism ensures high-precision positioning even in high-speed operations and under varying environmental conditions (such as temperature and vibration), effectively reducing the accumulation and propagation of errors and significantly improving the accuracy of the robot's execution.

[0039] 3. Support for nonlinear error correction and strong robustness: The present invention achieves preliminary calibration of arm length and zero offset by constructing an overdetermined homogeneous linear equation system and combining the Jacobi method to solve the eigenvector corresponding to the minimum eigenvalue. Then, through SVR error correction, it can adaptively handle complex nonlinear errors. Compared with traditional linear error models, the present invention has stronger robustness, especially when facing nonlinear errors such as robot mechanical wear and joint axis offset, it can still ensure high-precision operation and has wider adaptability.

[0040] 4. High degree of automation, improving industrial production efficiency: The present invention greatly reduces dependence on manual operations and reduces the interference of human errors through automated calibration and error compensation processes. Especially in modern industrial scenarios, such as electronic component assembly, precision machining and other occasions with extremely high requirements for positioning accuracy, the present invention can automatically adapt to different tasks and working environments, reduce downtime, and improve production efficiency. In particular, after the present invention introduces SVR model training and automatic compensation mechanism, the robot does not need to be recalibrated frequently and can operate stably for a long time, further reducing maintenance costs.

[0041] 5. Data-driven error compensation and strong model generalization ability: The present invention adopts a data-driven error compensation method and randomly collects actual error data from multiple points to train the SVR model, so that the model can cover the diverse error distribution in the workspace and ensure the generalization ability of the model.

[0042] Additional aspects and advantages of the present invention will be set forth in part in the description which follows and, in part, will be obvious from the description which follows, or may be learned by practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0044] Figure 1 This is the kinematics diagram of the SCARA robot of the present invention;

[0045] Figure 2 This is a schematic diagram of the calibration plate of the present invention;

[0046] Figure 3 It is a schematic diagram of the process of the present invention;

[0047] Figure 4 This is a schematic diagram of the calibration steps of the present invention;

[0048] Figure 5 It is a diagram of the calculation results of the present invention;

[0049] Figure 6 Table 1 Experimental data diagram of the present invention;

[0050] Figure 7 Table 2 is a graph showing the experimental results of the present invention. DETAILED DESCRIPTION

[0051] The following is a clear and complete description of the technical solutions in the embodiments of the present invention. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts are within the scope of protection of the present invention.

[0052] See also Figure 1 —7. In an embodiment of the present invention, a method for calibrating the arm length and zero point of a SCARA robot using support vector regression is provided, the method comprising the following steps:

[0053] Step 1: Establish the kinematic equations of the SCARA robot: Based on the structure of the upper and lower arms of the SCARA robot, define its kinematic model, where:

[0054] The length of the upper arm is L1, and the length of the lower arm is L2;

[0055] The first joint angle is θ1, and the second joint angle is θ2;

[0056] The position of the SCARA robot end in the workspace is determined by the following equation:

[0057]

[0058] Step 2: Calibration point selection and data collection: Align the calibration pin at the end of the SCARA robot with the preset points M and N on the calibration plate, record the joint angles θ1 and θ2 at different positions, and establish an error model for arm length and zero offset based on the measured angles;

[0059] Step 3: Preliminary solution based on overdetermined homogeneous linear equations: Based on the collected multiple calibration point positions and the corresponding joint angles, construct an overdetermined homogeneous linear equation system:

[0060] Ax=0

[0061] Among them, the matrix A is composed of the coordinates and angle information of each calibration point. By solving the eigenvector corresponding to the minimum eigenvalue of the equation group, the arm length error and zero offset value of the SCARA robot are preliminarily estimated;

[0062] Step 4: Construction and training of support vector regression (SVR) model:

[0063] Use multiple sets of data points randomly collected in the SCARA robot workspace to record the error between the actual position and the ideal position. The error is the residual error after calibration.

[0064] The error data set is divided into a training set and a test set, and regression analysis is performed using the support vector regression SVR model. The model uses the radial basis kernel function RBF, and its loss function is defined as:

[0065]

[0066] in, is the weight vector, is the regularization parameter, and is a slack variable, representing the prediction error;

[0067] Step 5, error compensation: Based on the training results of the SVR model, the position error of the SCARA robot is predicted in real time when the robot is moving, and the target position of the SCARA robot is compensated according to the predicted error value. The compensation formula is:

[0068]

[0069] in, is the error value predicted by the SVR model;

[0070] Step 6, calculation and adjustment: Based on the compensated SCARA robot end position, further adjust the arm length and zero point position to ensure high-precision operation of the robot in different working environments.

[0071] As a further solution of the present invention, the calibration points include at least two preset points in the workspace of the SCARA robot, and the calibration process is performed by aligning the calibration needle with the preset points in sequence and recording the joint angles.

[0072] As a further solution of the present invention, the overdetermined homogeneous linear equations are solved by the Jacobi method to obtain the eigenvector corresponding to the minimum eigenvalue, which is used to preliminarily calculate the arm length error and zero point offset.

[0073] As a further solution of the present invention, the support vector regression model uses a radial basis kernel function (RBF) to process the nonlinear error of the SCARA robot in the workspace.

[0074] As a further solution of the present invention, the data in the error data set is obtained by random sampling to ensure the diversity and representativeness of the data.

[0075] As a further solution of the present invention, the SVR model is trained by a sequential minimum optimization (SMO) algorithm to minimize the error and determine the optimal support vector.

[0076] Example 1:

[0077] The SCARA robot arm length and zero point calibration and error compensation method based on support vector regression of the present invention is particularly suitable for precision industrial scenarios, especially in high-precision, high-speed operations such as electronic product assembly and medical device manufacturing, and can significantly improve the robot's operating accuracy and stability. Taking the assembly of electronic components as an example, SCARA robots usually need to complete the assembly operations of multiple precision components in a very short time. Due to the high frequency of operations, complex movements, and the small size of the electronic components themselves, any slight positioning deviation during operation may lead to assembly failure or product quality problems, thereby having a significant impact on production efficiency and yield rate.

[0078] To address these issues, the present invention first calculates the initial error between the arm length and the zero point by establishing a kinematic model of the SCARA robot. The SCARA robot has two joints, with an upper arm length of L1 and a lower arm length of L2. The position of the end in the workspace is determined by the joint angles θ1 and θ2. Under ideal conditions, the theoretical position calculation formula for the robot end is:

[0079]

[0080] in, Indicates the deviation of the joint angle.

[0081] In actual applications, due to mechanical assembly errors during the manufacturing process, component wear after long-term use, and environmental changes (such as temperature fluctuations and vibration), the robot's joint angles and arm lengths may deviate from their theoretical values, causing zero-point position drift. This drift accumulates errors, affecting the accuracy of the entire production process.

[0082] To address this issue, the present invention first selects calibration points and collects data. A calibration pin is installed at the end of the SCARA robot. Multiple preset points M and N on a calibration plate are used to record the joint angles θ1 and θ2 at different positions. Using these calibration points, an overdetermined homogeneous linear system of equations is constructed. The Jacobi method is then used to solve the eigenvector corresponding to the minimum eigenvalue of this system, providing a preliminary estimate of the robot arm length error and zero offset. This process ensures that arm and forearm length deviations can be corrected during the initial calibration process, correcting the zero point position and laying the foundation for subsequent dynamic error compensation.

[0083] Existing calibration methods often struggle to cope with such complex operating environments. While laser ranging methods offer high-precision measurements, they require expensive equipment, have demanding environmental requirements, and are susceptible to external interference. Mechanical alignment methods are limited by manual operation, resulting in low calibration accuracy and a long time consumption, making them unsuitable for the demands of modern high-speed production lines. While mathematical modeling and simulation methods can simulate and calibrate, they struggle to handle complex nonlinear errors, are computationally complex, and rely on model accuracy.

[0084] To overcome these technical bottlenecks, this paper introduces a support vector regression (SVR) model to dynamically compensate for residual errors after calibration. Specifically, after the SCARA robot completes preliminary calibration, 200 sets of point data within the workspace are randomly collected, and the differences between the actual and ideal positions are recorded to form training and test sets. The support vector regression (SVR) model is used to perform regression analysis using the radial basis kernel (RBF) function to establish an error compensation model. The loss function is:

[0085]

[0086] By optimizing this objective function, the model can minimize the residual error in robot operation and significantly improve the calibration accuracy.

[0087] In actual use, when a SCARA robot performs an assembly task, its sensors collect the robot's motion parameters in real time, including characteristic data such as position, velocity, and acceleration. Through the SVR model, the system can dynamically predict the robot's error value in different environments and automatically compensate for the robot's end position based on the predicted error. The compensation formula is:

[0088]

[0089] in, is the error value predicted by the SVR model;

[0090] Application Example 1: During the assembly of high-precision electronic components, SCARA robots must repeatedly grasp and precisely place tiny components in a very short time. Due to long-term operation, wear of the joint bearings and ambient temperature fluctuations can cause the robot's actual position to deviate from the theoretical trajectory. Using the SVR model proposed in this paper, the system can automatically identify and correct these errors, ensuring that the robot achieves the expected accuracy every time it places components, avoiding assembly misalignment.

[0091] Application Example 2: In medical device production lines, SCARA robots are used to assemble precision components, requiring minimal deviation during operation. Traditional calibration methods struggle to adapt to these high-precision scenarios, especially when the robot operates at high speeds, generating significant inertial forces that can cause positional deviations. The dynamic error compensation mechanism of this invention enables the robot to adjust its zero point position in real time during operation, ensuring stable, high precision even at high speeds.

[0092] This invention not only adapts to complex error conditions in diverse environments but also ensures the robot's continued efficient operation in various industrial scenarios through dynamic compensation mechanisms, particularly in tasks involving high precision and high speed. Compared to existing technologies, this invention incorporates a support vector regression (SVR) model to adaptively handle nonlinear errors, significantly improving the robustness and accuracy of robot operation and reducing the long-term impact of factors such as assembly errors and wear. Through automated, data-driven calibration and error compensation, this invention significantly improves production line efficiency and product quality.

[0093] Example 2:

[0094] In an electronic equipment manufacturing workshop, SCARA robots are used for precision assembly operations within confined spaces. These tasks require the robots to grasp and assemble electronic components with high precision and efficiency within extremely limited operating space. Traditional SCARA robots often suffer from operational errors due to inaccurate arm length and zero-point calibration, particularly in confined operating environments, making them unable to meet the high-precision requirements. These errors often stem from geometric deviations during the robot's manufacturing and assembly process, as well as mechanical wear and tear from long-term use, which can cause zero-point position drift.

[0095] To address this issue, this paper proposes a method for SCARA robot arm length and zero point calibration and error compensation based on support vector regression (SVR). In precision assembly scenarios within confined spaces, robots must repeatedly perform pick-and-place operations between multiple preset points. First, using the present invention's calibration method, a calibration pin is installed at the end of the SCARA robot, sequentially aligned with preset points M and N on the calibration plate. Joint angle data at these calibration points is collected, and the robot arm length and zero point deviation are preliminarily calculated. This ensures that the length difference between the upper and lower arms and the zero point offset are corrected during basic calibration.

[0096] However, the calibration process can only partially eliminate static errors in robot assembly and manufacturing, but it cannot address dynamic errors during robot operation. Especially in confined spaces, any slight accumulated error can lead to inaccurate positioning of the robot's end-point, affecting the accuracy of its gripping and assembly. Therefore, this paper introduces a support vector regression (SVR) model to compensate for errors by collecting dynamic parameters of the robot's operation in real time. The system continuously collects data such as the end-point position, velocity, and acceleration during high-speed operation, and uses the SVR model to predict the robot's error distribution under specific conditions.

[0097] For example, during precision assembly within a confined space, a SCARA robot must grasp tiny electronic components and precisely place them on an assembly board. Due to the tight assembly area, the robot can gradually deviate from its originally programmed path during repeated operations due to the accumulation of small errors caused by environmental vibrations and temperature fluctuations. The SVR dynamic compensation model of the present invention allows the system to identify these deviations in real time and adjust the target position based on the predicted error values, ensuring accurate grasping and placement each time and preventing assembly defects caused by errors.

[0098] In this embodiment, the method of the present invention successfully addresses the problem of inaccurate robot operation caused by environmental constraints and accumulated errors in confined spaces. Through real-time error compensation, it ensures the robot's positioning accuracy during high-speed operation. Compared to traditional calibration methods, this method not only improves the robot's adaptability in complex environments but also effectively reduces error accumulation during operation, significantly improving product assembly consistency and production efficiency.

[0099] Example 3:

[0100] See also Figure 1 , Figure 1 This is the positive solution diagram of SCARA robot kinematics.

[0101] Since the calibration of the SCARA robot arm length and zero point is only related to the first and second axes, its kinematics can be simplified to a planar two-axis robot.

[0102] like Figure 1 As shown, the robot's base coordinate system xOy is established at the base coordinate origin O. Points O and A are the robot's revolute joints, point B is the robot's end, OA is the robot's upper arm, length L1, and AB is the robot's lower arm, length L2. θ1 is the angle between the x-axis and OA; θ2 is the angle between OA and AB; and Δθ2 is the angular deviation between the theoretical and actual values ​​of the robot's upper and lower arms. From this, the forward kinematic equations for the SCARA robot can be written as:

[0103]

[0104] In addition, it is stipulated that when θ2 is between 0° and 180°, the SCARA robot is on the left-hand side; when θ2 is between 0° and -180°, the SCARA robot is on the right-hand side. The theoretical values ​​of L1 and L2 are 100 mm, and the theoretical value of Δθ2 is 0°.

[0105] See also Figure 2 , Figure 2 This is a schematic diagram of the calibration plate, where the length between point M and point N is d, and the length of d is 150 mm.

[0106] See also Figure 3 , Figure 3 It is a schematic diagram of the process of the present invention.

[0107] The calibration steps for the SCARA robot's end calibration pin to measure the corresponding angles at two preset points on the calibration plate are as follows:

[0108] (1) The calibration pin is fixed at the end of the robot. Drag the robot from the left hand side so that the calibration pin coincides with point M and record the angle θ1 of the two joints at this time. (1) θ2 (1) .

[0109] (2) The calibration needle is fixed at the end of the robot. Drag the robot from the right side so that the calibration needle coincides with point M and record the angle θ1 of the two joints at this time. (2) 、 θ2 (2) .

[0110] (3) The calibration needle is fixed at the end of the robot. Drag the robot from the left hand side so that the calibration needle coincides with point N and record the angle θ1 of the two joints at this time. (3) 、 θ2 (3) .

[0111] (4) The calibration pin is fixed at the end of the robot. Drag the robot from the right side so that the calibration pin coincides with point N and record the angle θ1 of the two joints at this time. (4) 、 θ2 (4) .

[0112] See also Figure 4 , Figure 4 Schematic diagram of the calibration steps.

[0113] According to the measured angle and the kinematics of the SCARA robot, the equations for the arm length and zero offset are constructed. According to the measured angle at point M and the positive kinematics equation of the SCARA robot, we can obtain:

[0114]

[0115] According to the measured angle at point N and the SCARA robot forward kinematics equation, we can get:

[0116]

[0117] Connecting the above two equations and simplifying them, we can get:

[0118] By using methods such as variable substitution, we can simplify the overdetermined homogeneous linear equations to obtain the following:

[0119] make , then:

[0120] The problem of solving an overdetermined homogeneous linear system of equations is transformed into the problem of solving the eigenvalues ​​and corresponding eigenvectors of the corresponding real symmetric matrix, and the Jacobi method is used to find the eigenvector corresponding to its minimum eigenvalue.

[0121] Clearly, the resulting system of equations is an overdetermined homogeneous linear system, that is, Ax = 0, where A is an m×n matrix with full column rank and m > n. Given that Ax = 0 has an exact solution only if rank(A) < n, the above system of equations does not have an exact solution.

[0122] Since there is no exact solution, we can find a set of approximate solutions x + So that:

[0123]

[0124] At this point, in order to obtain a non-zero solution that satisfies the above equation, some restrictions need to be added so that the minimum x is obtained while satisfying the conditions. + If it is a solution to the equation A = 0, then kx is obviously also (k is an arbitrary scalar), so we can restrict . Then we can construct a constrained least squares problem:

[0125]

[0126] The unconstrained optimization can then be constructed via the Lagrange multiplier method:

[0127]

[0128] In order to obtain The extreme values ​​of x and Find the partial derivatives:

[0129]

[0130] make ,have to:

[0131]

[0132] It follows that in order to obtain The extreme value of x and Satisfy the above formula. It is easy to get that x and They are The eigenvalues ​​and eigenvectors of .

[0133] There are There are many eigenvalues ​​and eigenvectors, and they all satisfy the above formula. Substitution We can get:

[0134]

[0135] Obviously, the solution of the overdetermined homogeneous linear system is The eigenvector corresponding to the minimum eigenvalue.

[0136] From the previous point, we can simplify the overdetermined homogeneous linear equations and know that:

[0137]

[0138] but It must be a real symmetric matrix. The following describes the method for solving the eigenvector corresponding to the minimum eigenvalue of a real symmetric matrix.

[0139] For a real symmetric matrix A, there must be an orthogonal matrix Q such that Q T AQ=D. D is a diagonal matrix, the elements on the main diagonal are the eigenvalues ​​of matrix A, and each column of the orthogonal matrix Q corresponds to the eigenvector of the corresponding element on the main diagonal of the diagonal matrix D.

[0140] The Jacobi method uses plane rotation to perform similarity transformation on matrix A, transforming A into a diagonal matrix, thereby solving the eigenvalues ​​and eigenvectors. pq , is a unit matrix with cosφ in row p, column p, row q, column q, and -sinφ in row p and column q, and sinφ in row q and column p. For such a plane rotation matrix, it is not difficult to verify that it is an orthogonal matrix. Therefore, for vector x,U pq x is equivalent to rotating the plane defined by the pth and qth coordinate axes by φ degrees. Let the matrix A1 = U pq T AU pq It is a rotation matrix and also an orthogonal matrix, so in fact the matrix A1 is similar to the matrix A, so their eigenvalues ​​are the same.

[0141] Let the element in the i-th row and j-th column of matrix A1 be bij , the element in the i-th row and j-th column of matrix A is a ij (i=0,1,2,…,n−1, j=0,1,2,…,n-1). The following equation gives the operational relationship between the two matrix elements:

[0142]

[0143] The specific steps of Jacobi method for solving matrix eigenvalues ​​and eigenvectors are summarized as follows:

[0144] (1) Initialize the eigenvector to a diagonal matrix V, with the main diagonal elements being 1 and the other elements being 0

[0145] (2) Find the element a with the largest absolute value among the elements on the non-main diagonal of A. pq

[0146] (3) Usage , calculate the rotation matrix

[0147] (4) Calculate the matrix A1 and multiply the current matrix V by the rotation matrix to get the current feature matrix V

[0148] (5) If the maximum value of the non-main diagonal elements of the current iteration of matrix A is less than the given threshold, stop the calculation, otherwise execute the above process. When the calculation stops, the eigenvalue is the main diagonal element of matrix A, and the characteristic matrix is ​​matrix V

[0149] (6) Extract the eigenvector corresponding to the minimum eigenvalue

[0150] According to the solution, the zero offset value can be obtained, from which the actual arm length of the SCARA robot can be calculated.

[0151] Using the above Jacobi method, we can calculate .

[0152] according to So, the zero point offset is:

[0153] ,

[0154] The ratio of the length of the upper arm to the length of the lower arm is:

[0155] .

[0156] The actual arm length is:

[0157]

[0158] Similarly, the difference between the measured angle at point M and the measured angle at point N can be obtained. .

[0159] Then we have:

[0160]

[0161] Example 4:

[0162] The key point of the present invention is to transform the problem of solving an overdetermined homogeneous linear system of equations into the problem of solving the eigenvalues ​​and corresponding eigenvectors of the corresponding real symmetric matrix, and use the Jacobi method to find the eigenvector corresponding to its minimum eigenvalue.

[0163] The present invention uses the "derivation method" to deduce the solution of the overdetermined homogeneous linear equations. The eigenvector corresponding to the minimum eigenvalue. Compared with the "SVD decomposition method", the method used in the present invention is more concise and efficient.

[0164] The present invention uses the "Jacobi method" to solve the eigenvector corresponding to the minimum eigenvalue of a complex matrix. Compared with other methods such as bisection method and least squares method, the method used in the present invention has fewer iterations and more accurate calculation results.

[0165] The primary source of robot error is geometric error. This refers to errors in the relative position between the axis of the SCARA robot's upper and lower arm joints due to factors such as clearance errors in the assembly itself, low assembly precision, machining errors, and component wear caused by prolonged use. These errors are typically minimized by improving machining and assembly precision. However, due to the inherent machining precision of CNC machine tools and unavoidable human operator errors, this approach cannot completely eliminate them. Therefore, zero-point calibration is necessary to improve the accuracy of SCARA robots. This involves first identifying the key geometric error parameters and then determining their actual values ​​through kinematic calibration.

[0166] However, the method of obtaining the actual value of the parameters through kinematic calibration is also limited by the accuracy of the obtained data. Therefore, it is necessary to introduce a regression analysis method based on support vector machine (SVM) to compensate for the error of the calibrated data. The specific method is as follows:

[0167] 1. Data Collection: After the robot is calibrated, randomly select 200 points within the workspace. These points should cover the entire working area of ​​the robot to ensure a representative and diverse sample. The robot moves sequentially along these points, and each time it reaches a point, the difference between the robot's actual position and its ideal position is recorded, known as the residual position error. This error serves as sample data.

[0168] The sample data is divided into training samples and test samples. The ratio is usually 80-20: training samples: 160 sets of data for model training; test samples: 40 sets of data for model verification and evaluation.

[0169] 2. Feature extraction: Determine the features used for regression analysis, which may include the robot's position coordinates (such as X, Y, Z), the distance between the target point and the actual point, the speed and acceleration of the robot's movement, etc.

[0170] 3. Build a Support Vector Regression (SVR) model: The goal of SVR is to find a hyperplane such that the distance between most data points and the plane is within a certain range (the ε-insensitive interval) while minimizing the complexity of the model. Unlike traditional regression, SVR focuses on the support vectors, which are the data points closest to the hyperplane.

[0171] SVR uses an ε-insensitive loss function. For each data point, if the error between its predicted value and the true value is less than ε, no loss is included; otherwise, a penalty is imposed based on the absolute value of the error. This method can tolerate a certain amount of prediction error, thereby reducing overfitting.

[0172] Then, by introducing the Lagrange multiplier, the loss function is transformed into an optimization problem, the goal of which is to minimize the following function:

[0173]

[0174] in, is the weight vector, C is the regularization parameter, ξ i is the slack variable, which represents the prediction error at each point.

[0175] SVR uses kernel functions (such as linear kernels, polynomial kernels, or radial basis function kernels) to project data into a high-dimensional space to capture nonlinear relationships. The choice of kernel function is crucial to model performance. Selecting an appropriate kernel function (such as the radial basis function) is crucial to capturing the nonlinear characteristics of the data. SVR hyperparameters are set: the regularization parameter C and the tolerance error threshold ε. These parameters can be optimized through cross-validation.

[0176] Model training: Using 160 training samples, the Sequential Minimization Optimization (SMO) algorithm is used to minimize the loss function and find the optimal support vectors and weights. In this process, SVR learns how to map input features to output errors to achieve effective error compensation.

[0177] Model testing and evaluation: Validate the trained SVR model using 40 test samples. Input the test data and predict the corresponding error values. Evaluate the model's performance using metrics such as root mean square error (RMSE) and mean absolute error (MAE) to examine the difference between the predicted results and the actual position error.

[0178] Error compensation: Once the model has been tested and verified, the robot can use it to compensate for errors in future movements. During each movement, the current feature data is input, and the SVR model predicts the error in the current position. The robot's target position is adjusted accordingly to improve the accuracy of the movement.

[0179] Results Analysis: Analyze the effects before and after error compensation, compare the actual position after compensation with the ideal position, and evaluate the effectiveness of the compensation strategy. If necessary, further optimize the model or retrain it to adapt to different working environments or tasks.

[0180] Through the above steps, the SVR method can be effectively used to compensate for the position error of the robot and improve its accuracy and reliability in practical applications.

[0181] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations that come within the meaning and range of equivalents of the claims be embraced therein.

Claims

1. A SCARA robot arm length and zero point calibration method based on support vector regression, characterized in that: The method comprises the following steps: Step 1: Establish the kinematic equations of the SCARA robot: Based on the structure of the upper and lower arms of the SCARA robot, define its kinematic model, where: The length of the upper arm is L1, and the length of the lower arm is L2; The first joint angle is θ1, and the second joint angle is θ2; The position of the SCARA robot end in the workspace is determined by the following equation: Step 2: Calibration point selection and data collection: Align the calibration pin at the end of the SCARA robot with the preset points M and N on the calibration plate, record the joint angles θ1 and θ2 at different positions, and establish an error model for arm length and zero offset based on the measured angles; Step 3: Preliminary solution based on overdetermined homogeneous linear equations: Based on the collected multiple calibration point positions and the corresponding joint angles, construct an overdetermined homogeneous linear equation system: Ax=0 Among them, the matrix A is composed of the coordinates and angle information of each calibration point. By solving the eigenvector corresponding to the minimum eigenvalue of the equation group, the arm length error and zero offset value of the SCARA robot are preliminarily estimated; Step 4: Construction and training of support vector regression (SVR) model: Use multiple sets of data points randomly collected in the SCARA robot workspace to record the error between the actual position and the ideal position. The error is the residual error after calibration. The error data set is divided into a training set and a test set, and regression analysis is performed using the support vector regression SVR model. The model uses the radial basis kernel function RBF, and its loss function is defined as: in, is the weight vector, is the regularization parameter, and is the slack variable; Step 5, error compensation: Based on the training results of the SVR model, the position error of the SCARA robot is predicted in real time when the robot is moving, and the target position of the SCARA robot is compensated according to the predicted error value. The compensation formula is: in, is the error value predicted by the SVR model; Step 6, calculation and adjustment: Based on the compensated SCARA robot end position, further adjust the arm length and zero point position to ensure high-precision operation of the robot in different working environments.

2. The SCARA robot arm length and zero point calibration method based on support vector regression according to claim 1 is characterized in that: The calibration points include at least two preset points in the workspace of the SCARA robot. During the calibration process, the calibration needle is aligned with the preset points in sequence and the joint angles are recorded.

3. The SCARA robot arm length and zero point calibration method based on support vector regression according to claim 1 is characterized in that: The overdetermined homogeneous linear equations are solved by the Jacobi method to obtain the eigenvector corresponding to the minimum eigenvalue, which is used to preliminarily calculate the arm length error and zero point offset.

4. The SCARA robot arm length and zero point calibration method based on support vector regression according to claim 1 is characterized in that: The support vector regression model adopts radial basis kernel function (RBF) to process the nonlinear error of the SCARA robot in the workspace.

5. The SCARA robot arm length and zero point calibration method based on support vector regression according to claim 1 is characterized in that: The data in the error dataset are obtained by random sampling to ensure the diversity and representativeness of the data.

6. The SCARA robot arm length and zero point calibration method based on support vector regression according to claim 1 is characterized in that: The SVR model is trained by the Sequential Minimization (SMO) algorithm to minimize the error and determine the optimal support vector.

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