Parallel robot compensation method combining data driving with few-parameter geometric error model

Abstract

The invention relates to a parallel robot calibration method, aims to provide a parallel robot compensation method combining data driving with a few-parameter error model, and aims to solve the problems of difficulty in error modeling of a parallel robot with a complex structure and geometric errors and non-geometric errors of residual parts which cannot be identified after traditional kinematics calibration. According to the technical scheme, the method comprises the steps that 1, a closed-loop vector method is used for conducting kinematics analysis on the parallel robot of the complex structure to obtain an inverse kinematics model; 2, establishing a few-parameter geometric error model for the inverse kinematics model by adopting a vector differential method; 3, selecting a least square method optimization function to carry out parameter identification on the parameters of the small-parameter geometric error model; 4, carrying out a kinematics calibration experiment, and updating a parameter identification result to eliminate part of geometric errors; 5, historical data of a calibration experiment are collected, and a Gaussian process regression machine learning model is used for parameter selection and model training; and 6, carrying out prediction compensation on residual unrecognizable geometric errors and non-geometric errors after calibration.

Description

TECHNICAL FIELD

[0001] The application relates to a parallel robot calibration method, in particular to a parallel robot compensation method combining data driving and a few-parameter geometric error model. BACKGROUND

[0002] The parallel robot has significant advantages in the fields of precision machining, motion simulation and medical instruments due to its high stiffness, high bearing capacity and excellent positioning accuracy.

[0003] To ensure its motion accuracy, kinematic calibration becomes a key technical means. However, the traditional error model has several inherent limitations: first, the idealized assumption in the modeling process often deviates from the actual working condition; second, a single error model is difficult to cover all geometric error parameters; third, the multi-closed-loop structure of the parallel mechanism will introduce the coordination error of the passive joint. These factors result in the residual part of the unidentifiable geometric error even after kinematic calibration. In addition, due to the highly nonlinear and strong coupling characteristics of non-geometric errors, accurate modeling based on mathematical methods faces serious challenges.

[0004] Therefore, it is necessary to propose a parallel robot compensation method combining data driving and a few-parameter error model, and form a complete set of kinematic calibration methods suitable for any complex structure parallel robot.

[0005] At present, the research content of kinematic calibration of parallel robots is relatively rich. For example, CN118578383A proposes a five-axis parallel robot calibration method based on screw theory. This method analyzes the motion screw, constraint screw and transmission force screw characteristics of each branch chain, establishes a comprehensive model of stiffness and flexibility of the system, and realizes high-precision kinematic calibration. However, this method can only be applied to parallel mechanisms with a single configuration, and the application scene is limited. CN113580148A uses the equivalent motion chain principle to complete the kinematic parameter calibration of the parallel robot; however, this method has limited effect on the final accuracy improvement of the parallel robot. SUMMARY

[0006] The purpose of the present application is to overcome the deficiencies of the prior art, and to provide a parallel robot compensation method combining data driving and a few-parameter error model, aiming to solve the problems of difficult error modeling of complex structure parallel robots and residual unidentifiable geometric and non-geometric errors after traditional kinematic calibration.

[0007] To achieve the above purpose, the present application provides the following technical scheme:

[0008] A parallel robot compensation method combining data driving and a few-parameter geometric error model, the steps of which are as follows:

[0009] Step one: Kinematics analysis of complex structure parallel robot by closed-loop vector method to get inverse kinematics model;

[0010] Step two: Establishing the few-parameter geometric error model by vector differential method to the inverse kinematics model;

[0011] Step three: Selecting the least square method optimization function lsqnonlin to identify the parameters in the few-parameter geometric error model;

[0012] Step four: Kinematics calibration experiment, updating the parameter identification results to eliminate part of the geometric error;

[0013] Step five: Collecting calibration experiment historical data, using Gaussian process regression (GPR) machine learning model to select parameters and train the model;

[0014] Step six: Predicting and compensating the residual geometric error and non-geometric error that cannot be identified after calibration.

[0015] In some embodiments, according to step one, the specific way is:

[0016] The inverse kinematics of complex structure parallel robot is derived by closed-loop vector method. First, fixed coordinate system {O-xyz} and moving coordinate system {o-uvw} are established on the fixed platform and moving platform respectively. Second, the closed-loop vector equation of each branch position vector is established according to the specific structural characteristics of the parallel robot, which is expressed as:

[0017]

[0018] In the formula, l i is the position vector of the i-th branch of the parallel robot; P is the position vector of the fixed coordinate system origin O pointing to the moving coordinate system origin o; is the position vector of the moving coordinate system origin o pointing to the joint center of the i-th branch of the parallel robot connected with the moving platform; is the position vector of the fixed coordinate system origin O pointing to the joint center of the i-th branch of the parallel robot connected with the fixed platform (or the nearest); O R o is the rotation matrix from the moving coordinate system {o-uvw} to the fixed coordinate system {O-xyz}, which is expressed as:

[0019]

[0020] In the formula, R z,γ , R y,β and R x,αRz(γ) Ry(β) Rx(α) respectively represent rotation matrixes about z-axis rotating γ angle, about y-axis rotating β angle and about x-axis rotating α angle; s and c respectively represent sine function and cosine function; s α , s β , s γ represent sine function values of angles α, β, γ; c α , c β , c γ represent cosine function values of angles α, β, γ.

[0021] It is worth noting that the closed-loop vector equation can be adjusted according to different parallel robot configurations. Finally, according to the determined drive pair arrangement of the parallel robot, the coordinate expression of each position vector in the closed-loop vector equation can be expressed to solve the analytical expression of the inverse kinematics of the specific parallel robot.

[0022] In addition, for the parallel robot with an additional end effector installed at the end of the moving platform, the end point of the effector is represented as:

[0023] r = P + d (3) O R o d (3)

[0024] In the formula, r is the position vector of the end point of the effector in the fixed coordinate system, P is the position vector of the fixed coordinate system origin O pointing to the moving coordinate system origin o, and d is the position vector of the end point of the effector in the moving coordinate system.

[0025] By using the coordinate value of the end point of the effector r, the inverse solution of formula (3) is used to solve the end pose of the moving platform, which is substituted into the analytical expression of the inverse kinematics of the specific parallel robot to obtain the driving value of the moving pair required to move the end effector of the mechanism to the specified position. In summary, the complete inverse kinematics model of the complex structure parallel robot can be obtained as:

[0026] q = [q1...q i ] T = g(r, ε + Δε) (4)

[0027] In the formula, q represents the driving value of each branch of the mechanism, ε represents the nominal geometric parameters of the mechanism, and Δε represents the geometric error parameters of the mechanism.

[0028] In some embodiments, according to step two, the specific manner is as follows:

[0029] First, according to the determined closed-loop vector equation formula (1) of the parallel robot and the specific structural characteristics, the geometric error parameters affecting the positioning accuracy of the end of the parallel robot are determined. Second, linear perturbation is performed on both sides of the closed-loop vector equation, and the closed-loop vector error transfer equation is represented as:

[0030]

[0031] where u i is the unit direction vector of the i-th limb of the parallel robot.

[0032] It should be noted that the specific form of the differential variable is different according to the different structural characteristics of the parallel robot. On this basis, through the properties of the unit vector and the rotation matrix, and the differential derivation of the specific variable matrix in formula (7), the few-parameter geometric error model can be arranged as follows:

[0033] Δp = JΔε (15)

[0034] where Δp is the pose error of the end effector of the parallel robot, which is determined by the specific degree of freedom; J is the mapping Jacobian matrix between the geometric error parameters of the parallel robot and the pose error of the end effector of the moving platform.

[0035] In some embodiments, according to step three, the specific manner is as follows:

[0036] The parameter identification process of the geometric error parameters is represented as:

[0037] Δε = arg min||e * || (16)

[0038] where Δε represents the geometric error parameters, ||e * || represents the target value of the minimized error.

[0039] This minimization problem is a nonlinear least squares problem with an iterative relationship. The actual parameter error needs to be obtained through multiple identification iterations, and the iterative relationship is:

[0040] Δε r+1 = Δε r + (J T J) -1 J T e * (17)

[0041] where r represents the iteration number, J represents the error mapping matrix, and J T represents the transpose of the error mapping matrix. When ||e * || is less than a set value or the difference between Δε r+1 and Δε r is less than a set value, the iteration is terminated.

[0042] In some embodiments, according to step four, the specific manner is as follows:

[0043] When conducting kinematic calibration experiments, a reflective target ball or T-Mac needs to be installed at the end of the parallel robot's dynamic platform. The laser beam is emitted by the tracker host and the reflected signal is received by the detection camera to form a closed-loop feedback.

[0044] The present invention uses the method of fitting the coordinate system with the data points to obtain the translation vector and rotation matrix of the measurement coordinate system and the fixed coordinate system. The transformation relationship between the actuator end point of the parallel robot in the fixed coordinate system and the measurement coordinate system can be expressed as:

[0045] r=r t +R t l t (18)

[0046] Where r t and R t They represent the translation vector and rotation matrix of the measurement coordinate system relative to the fixed coordinate system, respectively. t Represents the position vector of the parallel robot actuator end point in the measurement coordinate system.

[0047] Finally, using the experimental data, the parameter identification process described in step 3 is carried out through the established low-parameter geometric error model formula (15), and the geometric error parameters Δε obtained from the identification results are added item by item to the nominal geometric parameter ε of the mechanism to obtain the new geometric error parameter ε * , complete parameter compensation.

[0048] In some embodiments, according to step five, the specific method is as follows:

[0049] First, the input data of the GPR model is determined as the calibrated end effector pose P * , P * The new geometric error parameter ε * It is deduced that the output data are respectively determined as the residual position error ΔPosition and posture error ΔPose after calibration. Secondly, by selecting the appropriate mean function and kernel function, the input x and output f(x) of the training data point are fitted into a Gaussian process. The input and output of the calibration point data to be predicted are defined as (x * ,y * ), and assuming that y * and f(x) obey the joint Gaussian distribution, which can be expressed as:

[0050]

[0051] Where μ f represents the mean of the f(x) part, μ y represents y * The mean of the parts, K ff=k(x,x) represents the covariance matrix between each point of f(x), K fy =k(x,x * ) represents the covariance matrix between f(x) and y * <k(x,y yy ) represents the covariance matrix between each point of y * <k(x,y * ) represents the covariance matrix between each point of y * .

[0052] Then through Bayesian inference, the posterior distribution probability is calculated by using the likelihood function to determine the expression of the output of the prediction point, which is represented as:

[0053]

[0054] In the formula, the mean is the output expression of the prediction point.

[0055] Finally, the model is determined to select the RBF kernel function, the polynomial mean function and the Gaussian likelihood function as the basis setting. The maximum likelihood estimation is selected for hyperparameter optimization and the number of hyperparameter optimization iterations is set to 20 times. The parameter selection and model training are completed.

[0056] In some embodiments, according to step six, the specific way is:

[0057] The input of the GPR model is the complex structure parallel robot kinematic calibration after the dynamic platform end effector pose p * , and the output is the two-norm of the remaining position error ΔPosition and the attitude error ΔPose after calibration, a total of 6 input characteristic quantities and 1 output characteristic quantity, which is represented as:

[0058]

[0059] In the formula, P * represents the end effector pose after the few-parameter geometric error calibration; ΔPosition represents the remaining position error; ΔPose represents the remaining attitude error; Δx, Δy and Δz represent the errors of the end effector in the x, y and z directions respectively; Δα, Δβ and Δγ represent the errors of the end effector in the α, β and γ rotation angles respectively.

[0060] Finally, the calibration point input data P * which needs to be predicted and compensated is substituted into formula (21) for calculation, and the prediction compensation of the residual error after calibration is completed.

[0061] Compared with the prior art, the beneficial effects of the present application are:

[0062] (1) The present application adopts a few geometric parameter method to model geometric error, and then carries out secondary compensation to improve the kinematic calibration accuracy, thereby providing a new solution for the error modeling of complex structure parallel robots.

[0063] (2) The present application proposes a parallel robot compensation method combining data driving with a few parameter error model, and verifies the effectiveness of the method. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 The figure is a flowchart of the kinematic calibration method of the present application.

[0065] Figure 2 The figure is a three-dimensional model of the 6-PSS parallel robot of embodiment 1 of the present application.

[0066] Figure 3 The figure is a structural diagram and coordinate system distribution diagram of the 6-PSS parallel robot of embodiment 1 of the present application.

[0067] Figure 4 The (A) figure in the figure is a top view diagram of the moving platform of the 6-PSS parallel robot of embodiment 1 of the present application.

[0068] Figure 4 The (B) figure in the figure is a top view diagram of the fixed platform of the 6-PSS parallel robot of embodiment 1 of the present application.

[0069] Figure 5 The figure is a parameter identification flowchart of embodiment 1 of the present application.

[0070] Figure 6 The figure is a kinematic calibration experiment diagram of embodiment 1 of the present application.

[0071] Figure 7 The (a) figure in the figure is a position error diagram before and after the kinematic calibration experiment of the 6-PSS parallel robot of embodiment 1 of the present application.

[0072] Figure 7 The (b) figure in the figure is an attitude error diagram before and after the kinematic calibration experiment of the 6-PSS parallel robot of embodiment 1 of the present application.

[0073] Figure 8 The (a) figure in the figure is a position prediction error diagram after training of the calibration experiment data under the GPR model of embodiment 1 of the present application.

[0074] Figure 8 The (b) figure in the figure is an attitude prediction error diagram after training of the calibration experiment data under the GPR model of embodiment 1 of the present application.

[0075] Figure 9 Figure (a) is a diagram showing the position error prediction compensation result after the calibration experiment under the GPR model of Example 1 of the present invention.

[0076] Figure 9 Figure (b) is a graph showing the attitude error prediction and compensation results after the calibration experiment under the GPR model of Example 1 of the present invention. DETAILED DESCRIPTION

[0077] The principle and specific implementation methods of the present invention are further described below with reference to the accompanying drawings.

[0078] Example 1 (6-PSS parallel robot calibration):

[0079] like Figure 2 As shown in the figure, the 6-PSS parallel robot (P: moving joint, S: spherical joint) consists of a moving platform, a fixed platform, and six PSS kinematic branches with identical topological structures. Each branch utilizes a linear guide-slider-connecting rod-spherical joint configuration: the connecting rod centerline is collinear with the line connecting the geometric centers of the upper and lower spherical joints; the slider is rigidly connected to the lower spherical joint via a base, while the upper spherical joint is fixed to the bottom of the moving platform. The six branches are divided into three groups based on spatial symmetry. Each group consists of two parallel linear guides with equal spacing. Each guide group is fixed to the fixed platform and uniformly distributed at 120° along the circumference with the fixed platform as the reference plane. The upper spherical joints and the upper spherical joints of the nearest branches in a different group can be further divided into three groups based on spatial symmetry. The centerlines of the upper spherical joints in each group are equidistant radially from the geometric center of the moving platform and are also uniformly distributed at 120°. In the initial state, the moving platform and the fixed platform are horizontally arranged. The 6-PSS parallel robot adjusts the end-stage pose of the moving platform by actuating the P joints along the linear guides.

[0080] like Figure 3 As shown in the figure, the motion joints of the 6-PSS parallel robot are defined as follows: in the i-th branch chain, the center of the upper spherical joint fixed to the moving platform is A i (i=1,2,3,4,5,6), the center of the lower ball joint fixed to the moving pair slider in the same branch chain is B i (i=1,2,3,4,5,6), the center of the moving sub-slider in the same branch chain is C i (i=1,2,3,4,5,6). Establish a fixed coordinate system {O-xyz} on the fixed platform, with the origin O at the geometric center of the fixed platform. The z-axis points perpendicular to the fixed platform and upward. The y-axis points parallel to linear guide rails 1 and 2 and toward the long side of the fixed platform. The x-axis's direction is determined by the right-hand rule. Establish a fixed coordinate system {o-uvw} on the moving platform, with the origin o at the geometric center of the moving platform. The w-axis points perpendicular to the moving platform and upward. The v-axis points parallel to the perpendicular bisector of the line connecting the upper spherical joints A1 and A2 and toward the short side of the moving platform. The u-axis's direction is determined by the right-hand rule.

[0081] As shown in Figure 4 is the top view of the moving platform and the fixed platform in the initial pose. In the moving platform, the upper spherical hinge is fixed to the moving platform through the base, and the centers of the six spherical hinges are uniformly arranged on the circumference of the same plane with a radius of r a , the central angle of the moving platform short side corresponding to the circle is , and the vertical distance between the spherical hinge center and the moving platform is h a . In the fixed platform, the parallel distance of the same group of linear guides is B1B2=B3B4=B5B6=l, and the vertical distance between the lower spherical hinge center and the fixed platform is h b . The length of the connecting rod connecting the upper and lower spherical hinges in each branch chain is L, the initial distance between the moving coordinate system and the fixed coordinate system origin is H, and the driving value of the moving pair slider of the ith branch chain is q i .

[0082] Step one: use the closed-loop vector method to analyze the kinematics of complex structure parallel robots to obtain the inverse kinematics model.

[0083] The closed-loop vector equation of each branch chain of the 6-PSS parallel robot can be expressed as:

[0084]

[0085] In the formula, the lower subscript i=(1:6), l i is the position vector of the moving coordinate system origin o pointing to the center of the joint connected with the moving platform in the ith branch chain of the parallel robot; is the position vector of the fixed coordinate system origin O pointing to the center of the joint connected with the fixed platform (or the closest) in the ith branch chain of the parallel robot; is the position vector of the fixed coordinate system origin O pointing to the center of the joint connected with the fixed platform (or the closest) in the ith branch chain of the parallel robot; O o R z,γ is the rotation matrix from the moving coordinate system {o-uvw} to the fixed coordinate system {O-xyz}, which is expressed as:

[0086]

[0087] In the formula, R y,β , R x,α and R z,γ , R y,β and R x,α respectively represent the rotation matrix about the z-axis by γ degrees, about the y-axis by β degrees and about the x-axis by α degrees; s and c respectively represent the sine function and the cosine function; s α , s β , s γ represent the sine function values of angles α, β, γ; c α , c β , c γcosine function value of angle α, β, γ.

[0088] 6-PSS parallel robot effector end point is expressed as:

[0089] r = P + d (3) O R o d (3)

[0090] In the formula, r is the position vector of the effector end point in the fixed coordinate system, P is the position vector of the fixed coordinate system origin O pointing to the moving coordinate system origin o, and d is the position vector of the effector end point in the moving coordinate system.

[0091] After obtaining the analytical expression of the effector end point, the inverse solution of the moving platform end pose (x, y, z, α, β, γ) is obtained, and then substituted into the inverse kinematics analytical expression. By calculating the driving pair value of the end effector at a specific position, a complete inverse kinematics model of the 6-PSS parallel robot is established

[0092] q = [q1... q i ] T = g(r, ε + Δε) (4)

[0093] In the formula, subscript i = (1:6), q represents the driving value of each branch of the mechanism, ε represents the nominal kinematics parameters of the mechanism, and Δε represents the corresponding kinematics error of the mechanism.

[0094] The robot driving values q1 to q6 are expressed as:

[0095]

[0096] In the formula, L represents the straight line distance A1B1, are the vectors of the upper spherical hinge center in the fixed coordinate system along the x, y, z axis direction. The expressions of omitted terms e3, e4, e5, e6 are as follows:

[0097]

[0098] Step 2: A few-parameter geometric error model is established for the inverse kinematics model using vector differential method.

[0099] The linear perturbation of the closed loop vector equation formula (1) of each branch chain of the 6-PSS parallel robot can be obtained as follows:

[0100]

[0101] Multiply the formula (7) by the unit vector and arrange it as:

[0102]

[0103] Ω = [αβγ] T represents the three rotational degrees of freedom of the moving platform.

[0104] The formula (8) can be rewritten as:

[0105] J s Δp = J e Δε (9)

[0106] In the formula, the matrix J s is the corresponding Jacobian matrix of the moving platform pose error Δp, which is expressed as:

[0107]

[0108] Δp is the moving platform pose error, which is expressed as:

[0109] Δp = [ΔPΔΩ] T = [ΔxΔyΔzΔαΔβΔγ] T (11)

[0110] The matrix J e is the corresponding Jacobian matrix of the branch chain link length error and the upper and lower spherical hinge center positioning error Δε e , which is expressed as:

[0111]

[0112] Δε e is the branch chain link length error and the upper and lower spherical hinge center positioning error, which is expressed as:

[0113]

[0114] In the formula, and can be expressed as:

[0115]

[0116] In summary, the few-parameter geometric error model between the geometric error parameters of the 6-PSS parallel robot and the pose error of the moving platform end is obtained as:

[0117] Δp = JΔε (15)

[0118] In the formula, J is the total Jacobian matrix,

[0119] Step three: select the least square optimization function lsqnonlin to identify the parameters in the few-parameter geometric error model.

[0120] The parameter identification process of geometric error parameters is represented as:

[0121] Δε = arg min ||e * || (16)

[0122] where Δε represents the set geometric error parameters, ||e * || represents the target value of minimized error.

[0123] The minimization problem is a nonlinear least square problem with iterative relationship. The actual parameter error needs to be obtained through multiple identification iterations, and the iterative relationship is:

[0124] Δε r+1 = Δε r + (J T J) -1 J T e * (17)

[0125] where r represents the iteration number, J represents the error mapping matrix, and J T represents the transpose of the error mapping matrix. When ||e * || is less than the set value or the difference between Δε r+1 and Δε r is less than the set value, the iteration is terminated.

[0126] The parameter identification process of the 6-PSS parallel robot is shown in Figure 5 When ||e * || is less than the set value or the difference between Δε r+1 and Δε r is less than the set value, the iteration is terminated. After the identified error parameters are compensated to the initial kinematic parameters, the kinematic forward calculation with the identified error parameters is performed, and the error-compensated moving platform pose information is obtained.

[0127] Step four: Perform kinematic calibration experiments to update the parameter identification results to eliminate part of the geometric error.

[0128] The kinematic calibration experiment of the 6-PSS parallel robot is shown in Figure 6 .

[0129] First, the laser receiving device of the tracker is installed at the end of the moving platform of the parallel robot, and the control system is used to drive the moving platform to move according to the given trajectory. Second, the real-time pose measurement of the mechanism end is performed by using the laser tracker (Leica-AT901-LR, Leica Geosystems AG) measurement system. Finally, the actual driving corresponding to each driving point is recorded by using the dSPACE software of the control system.

[0130] The present application adopts a method of fitting coordinate system with data points to obtain the translation vector and rotation matrix of the measurement coordinate system and the fixed coordinate system. The conversion relationship of the parallel robot actuator end point in the fixed coordinate system and the measurement coordinate system can be expressed as:

[0131] r = r t + R t l t (18)

[0132] In the formula, r t and R t respectively represent the translation vector and the rotation matrix of the measurement coordinate system relative to the fixed coordinate system, and l t represents the position vector of the parallel robot actuator end point in the measurement coordinate system.

[0133] Combined with the calibration experimental data, the parameter identification process described in step three is carried out through the established few-parameter geometric error model formula (15), and the geometric error parameters Δε obtained by the identification result are added to the nominal geometric parameters ε of the mechanism item by item to obtain the new geometric error parameters ε * , and the parameter compensation is completed.

[0134] As shown in Figure 7 (a), the average value of the two-norm of the position error vector is reduced from 5.7944 mm before calibration to 0.1868 mm. As shown in Figure 7 (b), the average value of the two-norm of the position and attitude error vector is reduced from 0.0143 rad before calibration to 0.00116 rad.

[0135] Step five: Collect calibration experimental historical data, and use Gaussian process regression (GPR) machine learning model for parameter selection and model training.

[0136] Firstly, the input data of the GPR model is determined as the end effector pose P * after calibration, and P * is derived from the new geometric error parameters ε * , and the output data is determined as the residual position error ΔPosition and attitude error ΔPose after calibration. Secondly, the input x and output f(x) of the training data points are fitted into a Gaussian process by selecting appropriate mean function and kernel function. The calibration point data to be predicted is input and output as (x * , y * ), and it is assumed that y * and f(x) obey the joint Gaussian distribution, which can be expressed as:

[0137]

[0138] In the formula, μ fμ represents the mean of f(x) part y y represents * K represents the mean of f(x) part ff k(x, x) represents the covariance matrix between each point of f(x), K fy k(x, x * ) represents the covariance matrix between f(x) and y * k(x yy , x * , x * ) represents the covariance matrix between each point of y * .

[0139] Then through Bayesian inference, the posterior distribution probability is calculated by using the likelihood function to determine the expression of the output of the prediction point, denoted as:

[0140]

[0141] In the formula, the mean is the output expression of the prediction point.

[0142] Finally, the model is determined to select the RBF kernel function, the polynomial mean function, and the Gaussian likelihood function as the basis setting. The maximum likelihood estimation is selected for hyperparameter optimization, and the number of hyperparameter optimization iterations is set to 20 times. The parameter selection and model training are completed.

[0143] The historical calibration data collected in the 6PSS kinematics calibration experiment is used to train the GPR model, and the prediction simulation result after training is as shown in Figure 8 . As shown in Figure 8 (a), when the output is the position error, the prediction error is 0.0129mm; as shown in Figure 8 (b), when the output is the attitude error, the prediction error is 0.000370 rad.

[0144] Step six: residual geometric error and non-geometric error that cannot be identified after calibration are predicted and compensated.

[0145] Through the GPR model, the error prediction compensation is carried out after the 6-PSS parallel robot calibration experiment. The input of the GPR model is the complex structure parallel robot kinematics calibration after the dynamic platform end effector pose p * , and the output is the two-norm of the residual position error ΔPosition and the attitude error ΔPose after calibration, a total of 6 input feature quantities and 1 output feature quantity, denoted as:

[0146]

[0147] In the formula, P *Position represents the remaining position error, and ΔPose represents the remaining pose error.

[0148] The input data P of the calibration points in the calibration experiment of the 6-PSS parallel robot * The prediction compensation results of the residual errors after calibration are calculated by substituting formula (21) as follows Figure 9 As shown in (a), the average value of the two-norm of the position error vector is reduced from 0.1868 mm before prediction to 0.1388 mm. As shown in (b), the average value of the two-norm of the pose error vector is reduced from 0.00116 rad before prediction to 0.000867 rad. Figure 9

[0149] In summary, the GPR model requires fewer training samples, which reduces the difficulty of collecting training data in the calibration experiment, and successfully realizes the secondary compensation of the kinematic calibration of the complex structure parallel robot, effectively improving the accuracy of the robot end effector.

[0150] The above examples only express the embodiments of the present application, and the description is more specific and detailed, but it cannot be understood as a limitation on the scope of the patent. It should be noted that for ordinary skilled in the art, without departing from the concept of the present application, a number of modifications and improvements can be made, which are within the scope of the present application. Therefore, the scope of the patent of the present application should be subject to the appended claims.

[0151] Although the embodiments of the present application have been shown and described, it can be understood by those skilled in the art that various changes, modifications, replacements and variations can be made to these embodiments without departing from the principles and spirits of the present application, the scope of the present application is defined by the appended claims and their equivalents.​

Claims

1. A data-driven parallel robot compensation method combined with a low-parameter geometric error model, the steps of which are as follows: Step 1: Use the closed-loop vector method to perform kinematic analysis on the complex structure parallel robot to obtain the inverse kinematics model; Step 2: Use vector differential method to establish a low-parameter geometric error model for the inverse kinematics model; Step 3: Select the least squares optimization function lsqnonlin to perform parameter identification on the parameters of the few-parameter geometric error model; Step 4: Perform kinematic calibration experiments and update the parameter identification results to eliminate some geometric errors; Step 5: Collect historical data of calibration experiments and use Gaussian process regression (GPR) machine learning model for parameter selection and model training; Step 6: Predict and compensate for the unidentifiable geometric errors and non-geometric errors remaining after calibration.

2. The data-driven parallel robot compensation method combined with a low-parameter geometric error model according to claim 1 is characterized by: The specific method of step one is: The closed-loop vector method is used to derive the inverse kinematic solution of the complex structure parallel robot. First, the fixed coordinate system {O-xyz} and the moving coordinate system {o-uvw} are established on the fixed platform and the moving platform, respectively. Second, the closed-loop vector equation of the position vector of each branch is established according to the specific structural characteristics of the parallel robot, which is expressed as: Where, l i is the sum of the position vectors of all links on the i-th branch of the parallel robot; P is the position vector from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; is the position vector of the origin o of the motion coordinate system pointing to the center of the joint connected to the moving platform in the i-th branch of the parallel robot; The position vector of the fixed coordinate system origin O pointing to the center of the joint in the i-th branch of the parallel robot that is connected to (or closest to) the fixed platform; O R o is the rotation matrix from the moving coordinate system {o-uvw} to the fixed coordinate system {O-xyz}, expressed as: Where R z,γ , R y,β and R x,α They represent the rotation matrices for the rotation of γ angle about the z-axis, β angle about the y-axis, and α angle about the x-axis; s and c represent the sine function and cosine function respectively; s α , s β , s γ Represents the sine function value of angles α, β, and γ; c α , c β , c γ Represents the cosine function value of angles α, β, and γ; The closed-loop vector equation is adjusted differently according to the configuration of the parallel robot. According to the determined drive pair arrangement of the parallel robot, the coordinate expression of each position vector in the closed-loop vector equation can be expressed to obtain the analytical expression of the inverse kinematic solution of the specific parallel robot. For a parallel robot with an additional end effector installed at the end of the moving platform, the end point of the actuator is expressed as: r=P+ O R o d (3) Where r is the position vector of the actuator end point in the fixed coordinate system, P is the position vector from the origin O of the fixed coordinate system to the origin o of the moving coordinate system, and d is the position vector of the actuator end point in the moving coordinate system. By using the coordinate value of the actuator end point r, the end position of the moving platform is inversely solved using formula (3). Substituting it into the analytical expression of the specific parallel robot kinematic inverse solution, the required driving value of the moving pair required to move the end point of the mechanism end effector to the specified position can be obtained. The complete inverse kinematics model of the complex structure parallel robot can be obtained as follows: q=[q1...q i ] T =g(r,ε+Δε) (4) Where q represents the driving value of each branch of the mechanism, ε represents the nominal geometric parameter of the mechanism, and Δε represents the geometric error parameter of the mechanism.

3. The data-driven parallel robot compensation method combined with a low-parameter geometric error model according to claim 2 is characterized by: The specific method of step 2 is: First, according to the determined closed-loop vector equation formula (1) of the parallel robot and the specific structural characteristics, the geometric error parameters that affect the positioning accuracy of the end of the parallel robot are determined; secondly, linear perturbations are performed on both sides of the closed-loop vector equation, and the closed-loop vector error transfer equation is expressed as: Where u i is the unit direction vector of the i-th branch link of the parallel robot; According to the different structural characteristics of the parallel robot, the specific differential variable form is different; through the properties of the unit vector and the rotation matrix, as well as the differential derivation of the specific variable matrix in formula (7), the low-parameter geometric error model can be organized into the following form: Δp=JΔε (15) Where Δp is the pose error of the end effector of the parallel robot, which is determined by the specific degree of freedom; J is the mapping Jacobian matrix between the geometric error parameters of the parallel robot and the pose error of the end effector of the moving platform.

4. The data-driven parallel robot compensation method combined with a low-parameter geometric error model according to claim 3 is characterized by: The specific method of step three is: The parameter identification process of the geometric error parameters is expressed as: Δε=arg min||e * || (16) Where Δε represents the geometric error parameter, ||e * || represents the target value for minimizing the error; The minimization problem is a nonlinear least squares problem with an iterative relation; The actual parameter error can only be obtained after multiple identification iterations, and its iterative relationship is: No r+1 =No r +(J T J) -1 J T e * (17) Where r represents the number of iterations, J represents the error mapping matrix, and J T represents the transpose of the error mapping matrix; when ||e * ||Less than the set value or Δε r+1 and Δε r The iteration is terminated when the difference is less than the set value.

5. The data-driven parallel robot compensation method combined with a low-parameter geometric error model according to claim 4 is characterized in that: The specific method of step 4 is: When conducting kinematic calibration experiments, a reflective target ball or T-Mac needs to be installed at the end of the parallel robot's dynamic platform. The tracker host emits a laser beam and the detection camera receives the reflected signal to form a closed-loop feedback loop. The method of fitting the coordinate system with the data points is used to obtain the translation vector and rotation matrix of the measurement coordinate system and the fixed coordinate system. The transformation relationship between the end point of the parallel robot actuator in the fixed coordinate system and the measurement coordinate system can be expressed as follows: r＝r t +R t l t (18) Where r t and R t They represent the translation vector and rotation matrix of the measurement coordinate system relative to the fixed coordinate system, respectively. t Represents the position vector of the end point of the parallel robot actuator in the measurement coordinate system; Finally, using the experimental data, the parameter identification process described in step 3 is carried out through the established low-parameter geometric error model formula (15), and the geometric error parameters Δε obtained from the identification results are added item by item to the nominal geometric parameter ε of the mechanism to obtain the new geometric error parameter ε * , complete parameter compensation.

6. The data-driven parallel robot compensation method combined with a low-parameter geometric error model according to claim 5 is characterized by: The specific method of step five is: First, the input data of the GPR model is determined as the calibrated end effector pose P * , P * The new geometric error parameter ε * It is inferred that the output data are respectively determined as the residual position error ΔPosition and posture error ΔPose after calibration; secondly, by selecting the appropriate mean function and kernel function, the input x and output f(x) of the training data point are fitted into a Gaussian process; the input and output of the calibration point data to be predicted are defined as (x * ,y * ), and assuming that y * and f(x) obey the joint Gaussian distribution, which can be expressed as: Where μ f represents the mean of the f(x) part, μ y represents y * The mean of the parts, K ff =k(x,x) represents the covariance matrix between the points of f(x), K fy =k(x,x * ) represents the relationship between f(x) and y * The covariance matrix between yy =k(x * ,x * ) represents y * The covariance matrix between each point; Then, through Bayesian inference, the likelihood function is used to calculate the posterior distribution probability to determine the expression of the predicted point output, which is expressed as: In the formula, the mean That is the output expression of the prediction point; Finally, the model was determined to select the RBF kernel function, polynomial mean function, and Gaussian likelihood function as the basic settings; maximum likelihood estimation was selected for hyperparameter optimization and the number of hyperparameter optimization iterations was set to 20; parameter selection and model training were completed.

7. The data-driven parallel robot compensation method combined with a low-parameter geometric error model according to claim 6 is characterized in that: The specific method of step six is: The input of the GPR model is the position p of the end effector of the dynamic platform after kinematic calibration of the complex structure parallel robot. * , the outputs are the two norms of the residual position error ΔPosition and the posture error ΔPose after calibration, a total of 6 input feature quantities and 1 output feature quantity, expressed as: Where, P * represents the pose of the end effector after calibration of the low-parameter geometric error, ΔPosition represents the residual position error, and ΔPose represents the residual pose error; Δx, Δy, and Δz represent the errors of the end effector in the x, y, and z directions, respectively; Δα, Δβ, and Δγ represent the errors of the end effector in the α, β, and γ rotation angles, respectively; Finally, the calibration point that needs to be predicted and compensated is input into the data P * Substituting into formula (21) to calculate, the prediction compensation of the residual error after calibration is completed.

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