Error compensation method for 2pru-psr parallel robot

Abstract

The present application relates to a kind of error compensation methods, the purpose is to provide a kind of 2PRU-PSR parallel robot error compensation method, this method has the characteristics of low cost, high efficiency and real-time nature.The technical scheme is a kind of 2PRU-PSR parallel robot error compensation method, according to the following steps:step one: using closed loop vector method to establish the inverse kinematics model of 2PRU-PSR parallel robot and using Newton iteration formula to calculate to obtain complete forward kinematics model;Step two: set error parameter, utilize vector differential method to construct 2PRU-PSR parallel robot error mapping model;Step three: complete kinematic parameter identification and calibrate simulation to verify the correctness of the model built;Step four: using the nonparametric model based on neural network to the end of 2PRU-PSR parallel robot after kinematic calibration remaining error is predicted, using nonparametric model to predict error to complete compensation.

Description

Technical Field

[0001] This invention relates to an error compensation method, specifically an error compensation method for a 2PRU-PSR parallel robot that combines parametric and non-parametric models. Background Technology

[0002] Parallel robots are characterized by high precision, high rigidity, and high load-bearing capacity, making them well-suited for applications in precision machining. Currently, the sources of error affecting robot positioning accuracy can be categorized into two types: geometric parameter errors and non-geometric parameter errors.

[0003] Geometric parameter error compensation for parallel robots typically employs kinematic calibration methods, which generally include geometric error modeling, parameter identification, error measurement, and error compensation. However, most current geometric error modeling methods struggle to establish error models that simultaneously possess completeness, continuity, and minimization. Furthermore, parallel robots still exhibit certain pose errors after kinematic calibration, and the kinematic calibration process neglects the impact of non-geometric parameter errors on the accuracy of parallel robots.

[0004] However, non-geometric error parameters are highly nonlinear and strongly coupled, so modeling and calibration methods for non-geometric error parameters are not universally applicable. Therefore, some scholars have proposed parameter-free model calibration methods for non-geometric error parameters that do not require analytical error models or specific parameter identification methods.

[0005] For improving the accuracy of parallel robots, error compensation solely through kinematic calibration is insufficient in precision. Furthermore, it's difficult to establish accurate error parameter models for the remaining, unidentifiable geometric and non-geometric errors after calibration. The highly nonlinear nature of non-geometric errors makes modeling challenging. Employing a parameter-free model and utilizing the nonlinear fitting capabilities of neural networks for prediction and compensation not only solves the modeling difficulty but also improves the real-time performance of error compensation. However, directly using neural networks to predict and compensate for the impact of geometric parameter errors increases the uncertainty of accuracy improvement and reduces stability. Summary of the Invention

[0006] The purpose of this invention is to address the aforementioned background technology by proposing a 2PRU-PSR parallel robot error compensation method, which is characterized by low cost, high efficiency, and real-time performance.

[0007] The technical solution provided by this invention is:

[0008] A method for error compensation in a 2PRU-PSR parallel robot is described, comprising the following steps:

[0009] Step 1: Establish the inverse kinematics model of the 2PRU-PSR parallel robot using the closed-loop vector method and calculate the complete forward kinematics model using Newton's iterative formula;

[0010] Step 2: Set error parameters and construct the 2PRU-PSR parallel robot error mapping model using the vector differential method;

[0011] Step 3: Complete the kinematic parameter identification and perform calibration simulation to verify the correctness of the built model;

[0012] Step 4: Use a non-parametric model based on neural networks to predict the residual error at the end of the 2PRU-PSR parallel robot after kinematic calibration, and use the non-parametric model to predict the error to complete the compensation.

[0013] In step one, for the 2PRU-PSR parallel mechanism, the inverse kinematic model is expressed as:

[0014] q = [q1 q2 q3] T =g(r,ε+Δε) (1)

[0015] In the formula, q represents the total driving value of the mechanism; q n (n=1,2,3) represents the driving value of the moving pair on branch n; T represents the inverse solution operation of the matrix; g represents the functional relationship; r represents the position of the end effector of the mechanism; ε represents the nominal kinematic parameter of the mechanism; Δε represents the kinematic error of the mechanism; g(r,ε+Δε) represents the functional relationship with r and ε+Δε as parameters.

[0016] The inverse kinematics expressions for the three moving pair drive values ​​(q1, q2, q3) are shown below:

[0017]

[0018] In the formula: α and β represent the output angles of the moving platform, z0 represents the z-axis coordinate of the moving platform, and the radius of the circumscribed circle of the moving platform is oB. n =lb n (n = 1, 2, 3), and lb1 = lb2 = e1, lb3 = e2; s, c, and t represent the abbreviations of the sine, cosine, and tangent functions, respectively; s α c represents the sine value of α. α t represents the cosine value of α. α s represents the tangent of α; β c represents the sine value of β. β t represents the cosine value of β. β Represents the tangent of β; l n (n = 1, 2, 3) represents link A n B nThe length of ; R represents the rotation matrix.

[0019] Newton's iteration method is chosen to obtain the end-effector pose information and end-effector position information corresponding to the driving input. The Newton's iteration formula is expressed as:

[0020]

[0021] In the formula, x n Let x represent the nth iteration. n+1 f'(x) represents the (n+1)th iteration; n ) is f(x) n The derivative of ).

[0022] The inverse kinematics expression of the moving pair drive is written in matrix form. Then, the derivative matrix is ​​obtained by successively differentiating the pose information (α, β, z0) of the moving platform in the inverse matrix. The derivative matrix is ​​then substituted into Newton's iteration formula to set the initial value of the iteration and the accuracy condition for the termination of the cycle, thereby obtaining the complete forward kinematics model of the 2PRU-PSR parallel mechanism with the end effector:

[0023] r = [r x r y r z ] T =f(q,ε+Δε) (3)

[0024] In the formula: r represents the position of the end effector of the mechanism; r n (n = x, y, z) represent the components of r on the x, y, and z axes, respectively; f(q, ε + Δε) represents the functional relationship with r and ε + Δε as parameters.

[0025] In step two, the position coordinates of the end effector r can be expressed as:

[0026] r = p + O R o d (4)

[0027] In the formula: p represents the position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; O R O d represents the rotation matrix from the moving coordinate system to the fixed coordinate system; d represents the coordinates of the end effector position of the mechanism in the moving platform coordinate system.

[0028] Where: the position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system can be represented as:

[0029] p = [0 y0 z0] T (5)

[0030] Where y0 represents the y-axis coordinate of the moving platform. Specifically:

[0031]

[0032] In the formula: l an (n = 1, 2, 3) represents the radius of the circumcircle of the static plateau on branch n, l bn (n = 1, 2, 3) represents the radius of the circumcircle of the moving platform on branch n; l n (n = 1, 2, 3) represents link A on branch n. n B n The length.

[0033] Applying linear perturbation to both sides of equation (4) yields:

[0034] Δr=Δp+Δ O R o d+ O R o Δd=JΔε (6)

[0035] △r represents the position error of the end effector of the mechanism; △p represents the position vector error from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; △ O R o Δd represents the rotation matrix error from the moving coordinate system to the fixed coordinate system; Δd represents the coordinate error of the end effector position in the moving platform coordinate system; J represents the error mapping Jacobian matrix and kinematic error parameters.

[0036] Linear perturbation of equation (5) and its matrix multiplication form is as follows:

[0037]

[0038] Among them: J p The matrix represents the Jacobian matrix of the position vector p after linear perturbation with respect to the attitude error of the moving platform's end effector; △α and △β represent the output angle error of the moving platform, and △z0 represents the z-axis coordinate error of the moving platform; and J p1 J p2 They are respectively:

[0039]

[0040]

[0041] According to the differential formula for the rotation matrix, the rotation matrix ΔR can be expressed as:

[0042] ΔR=δΩ×R=S(δΩ)R (8)

[0043] Where: △R represents the rotation matrix error; δΩ represents the attitude deviation of the moving platform of the mechanism; R represents the rotation matrix; S(δΩ) is the antisymmetric matrix of δΩ.

[0044] Solve for the antisymmetric matrix S(δΩ) in equation (8), and then solve for the vector δΩ using the properties of the antisymmetric matrix, and write it in the form of matrix multiplication:

[0045]

[0046] Among them: J wp Jacobian mapping matrix representing the deviation of the moving platform of the mechanism.

[0047] Substituting equations (7) and (9) into equation (6), we get:

[0048]

[0049] Where: S(Rd) represents the antisymmetric matrix of the product of Rd.

[0050] In equation (19), q i Moving (i = 1, 2, 3) to the right side of the equation and substituting the signs from the error modeling part, we get three equations (equ1, equ2, equ3). Then, performing total differential on both sides of the equations yields:

[0051]

[0052] Among them: A n (n = 1, 2, 3) represents the unknown, B n (n = 1, 2, 3) represents the unknowns, C n (n = 1, 2, 3) represents the unknown, D n (n = 1, 2, 3) represents the unknown, E n (n = 1, 2, 3) represents the unknown, F n (n = 1, 2, 3) represents the unknown; △l an (n=1,2,3) represents the error of the circumcircle radius of the static plateau on branch n, Δl bn (n=1,2,3) represents the error of the circumscribed circle radius of the moving platform on branch n; l n (n = 1, 2, 3) represents link A on branch n. n B n The length.

[0053]

[0054] In the formula: Representing equ i For l ai Find the partial derivative; Representing equ i For l bi Find the partial derivative; Represents equ iFor l i Find the partial derivative; Represents equ i Take the partial derivative with respect to α; Represents equ i Take the partial derivative with respect to β; Represents equ i Take the partial derivative with respect to z0; This represents integral operations.

[0055] Taking (Δα, Δβ, Δz0) as the unknown quantity, Δε 0 As the parameters of the equation, equation (11) is expressed in matrix form:

[0056]

[0057] in:

[0058]

[0059]

[0060] Δl a =[δl a1 δl a2 δl a3 ] T

[0061] Δl b =[δl b1 δl b2 δl b3 ] T

[0062] Δl=[δl1 δl2 δl3] T

[0063] In the formula: δl an (n = 1, 2, 3) represents the radius of the circumcircle of the static platform; δl bn (n = 1, 2, 3) represents the radius of the circumcircle of the moving platform; δl n (n = 1, 2, 3) represents the length Δl of the branch link. a This represents the error in the radius of the circumscribed circle of the static platform; Δl b This represents the error in the circumscribed circle radius of the moving platform; Δl n This represents the error in the length of the branch link.

[0064] This establishes the mapping relationship between the kinematic error of the mechanism and the end-effector attitude error of the moving platform:

[0065] [Δα Δβ Δz0] T =J e Δε 0 (14)

[0066] In the formula: J e The Jacobian matrix represents the relationship between the residual kinematic error of the mechanism and the end-effector attitude error of the moving platform; Δε 0 Represents the remaining kinematic error parameters and:

[0067]

[0068] Substituting equation (14) into equation (10) yields the final error mapping model for the 2PRU-PSR mechanism as follows:

[0069] Δr=JΔε (15)

[0070] In the formula: J represents the error mapping Jacobian matrix and the kinematic error parameters, specifically:

[0071] J = [J q R],Δε=[Δε 0 Δd] T

[0072] J q The matrix is ​​a partial Jacobian matrix expression in matrix form after simplification of equation (10), as follows:

[0073] J q =J p J e -S(Rd)J wp J e (16)

[0074] In step three, the least squares sum of squares optimization method is selected for identification calculation. The attitude error and position error before and after kinematic calibration are compared to verify the correctness of the built model.

[0075] In step four, a BP (Back Propagation) neural network is selected to predict the remaining localization error. The predicted network topology is determined to be a BP neural network model consisting of an input layer, a hidden layer, and an output layer.

[0076] Before using the network for error prediction, the network model needs to be trained with a large amount of data. The network is configured with 6 inputs and 1 output, and the purelin function is selected as the network transfer function. Gradient training is used. The maximum number of training iterations is set to 5000, the learning rate is 0.01, the minimum allowable error is set to 0.0001, and the weights and thresholds are randomly set to 1 or 2.

[0077] Referring to the following empirical formula, different numbers of hidden layer nodes can be selected to train the 2PRU-PSR mechanism on data samples.

[0078]

[0079] In the formula: p represents the number of hidden layer nodes, m and n represent the number of input layer and output layer nodes, and a is an adjustment parameter ranging from 1 to 10.

[0080] Under the same training sample size and learning rate, different numbers of hidden layer nodes were selected for step-by-step experiments, and prediction simulations were performed after the network model was trained. Finally, it was determined that the number of hidden layer nodes should be 10 when training pose error data and training position error data.

[0081] Training was conducted with different numbers of samples to find the best performance. Then, the prediction error of the neural network-based non-parametric model was compensated after kinematic calibration of the 2PRU-PSR parallel robot.

[0082] Compared to existing driver selection technologies, the advantages of this invention are as follows:

[0083] (1) Significant results. The residual error at the end of the 2PRU-PSR parallel robot after kinematic calibration is predicted by a non-parametric model based on neural networks, and secondary compensation is performed after prediction. This provides an important theoretical basis and reference for error compensation of many parallel mechanisms that still have large residual errors after kinematic calibration alone.

[0084] (2) Low cost and high efficiency;

[0085] (3) It has real-time capability. Attached Figure Description

[0086] Figure 1 This is a flowchart of the kinematic calibration simulation of the mechanism according to an embodiment of the present invention.

[0087] Figure 2 This is a flowchart of the BP neural network prediction error compensation process according to an embodiment of the present invention.

[0088] Figure 3 This is a schematic diagram of the BP neural network structure according to an embodiment of the present invention.

[0089] Figure 4 This is a schematic diagram of the coordinate system of the 2PRU-PSR parallel mechanism in Embodiment 1 of the present invention.

[0090] Figure 5 This is a comparison diagram of the kinematic errors before and after calibration of the 2PRU-PSR mechanism in Embodiment 1 of the present invention.

[0091] Figure 6 This is a comparison chart of the error before and after the prediction error compensation of the 2PRU-PSR mechanism based on the parameterless model of the neural network in Embodiment 1 of the present invention. Detailed Implementation

[0092] The present invention will be further described below with reference to the embodiments shown in the accompanying drawings.

[0093] The present invention proposes an error prediction and compensation method for 2PRU-PSR parallel robots based on a combination of parametric and non-parametric models. First, a kinematic model and an error mapping model of the 2PRU-PSR parallel robot are established; second, the kinematic parameters are identified; and finally, error prediction and compensation are performed based on a neural network algorithm using both the non-parametric and non-parametric models.

[0094] Specifically, the present invention consists of the following four steps (e.g. Figure 1-3 ):

[0095] In step one, for the 2PRU-PSR parallel mechanism, the inverse kinematic model is expressed as:

[0096] q = [q1 q2 q3] T =g(r,ε+Δε) (1)

[0097] In the formula, q represents the total driving value of the mechanism; q n (n=1,2,3) represents the driving value of branch n; T represents the inverse solution operation of the matrix; g(q,ε+Δε) represents the functional relationship with r and ε+Δε as parameters; r represents the position of the end effector of the mechanism; ε represents the nominal kinematic parameters of the mechanism; Δε represents the kinematic error of the mechanism.

[0098] Newton's iteration method is chosen to obtain the end-effector pose information and end-effector position information corresponding to the driving input; the Newton iteration formula is expressed as:

[0099]

[0100] In the formula, x n Let x represent the nth iteration. n+1 f'(x) represents the (n+1)th iteration; n ) is f(x) n The derivative of ).

[0101] The inverse kinematics expression of the moving pair drive is written in matrix form. Then, the derivative matrix is ​​obtained by successively differentiating the pose information (α, β, z0) of the moving platform in the inverse matrix. Substituting these derivative matrices into Newton's iteration formula to set the initial values ​​of the iteration and the accuracy conditions for the termination of the cycle, a complete forward kinematics model of the 2PRU-PSR parallel mechanism with an end effector is obtained.

[0102] r = [r x r y r z ] T =f(q,ε+Δε) (3)

[0103] In the formula: r represents the position of the end effector of the mechanism; r n (n = x, y, z) represent the components of r on the x, y, and z axes, respectively; f(q, ε + Δε) represents the functional relationship with r and ε + Δε as parameters.

[0104] In step two, the position r of the end effector of the mechanism is represented by the following coordinates:

[0105] r = p + O R o d (4)

[0106] In the formula, p is the position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; o R o d is the rotation matrix from the moving coordinate system to the fixed coordinate system; d is the coordinate of the end effector position of the mechanism in the moving platform coordinate system.

[0107] Where: the position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system can be represented as:

[0108] p = [0 y0 z0] T (5)

[0109] Where y0 represents the y-axis coordinate of the moving platform. Specifically:

[0110]

[0111] In the formula: l an (n = 1, 2, 3) represents the radius of the circumcircle of the static plateau on branch n, l bn (n = 1, 2, 3) represents the radius of the circumcircle of the moving platform on branch n; l n (n = 1, 2, 3) represents link A n B n The length.

[0112] Applying linear perturbation to both sides of equation (4) yields:

[0113] Δr=Δp+Δ O R o d+ O R o Δd=JΔε (6)

[0114] △r represents the position error of the end effector of the mechanism; △p represents the position vector error from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; △ O R O Δd represents the rotation matrix error from the moving coordinate system to the fixed coordinate system; Δd represents the coordinate error of the end effector position in the moving platform coordinate system.

[0115] Linear perturbation of equation (5) and its matrix multiplication form is as follows:

[0116]

[0117] Among them: J p The matrix represents the Jacobian matrix of the position vector p after linear perturbation with respect to the attitude error of the moving platform's end effector; △α and △β represent the output angle error of the moving platform, and △z0 represents the z-axis coordinate error of the moving platform; and J p1 J p2 They are respectively:

[0118]

[0119]

[0120] According to the differential formula for the rotation matrix, the rotation matrix ΔR can be expressed as:

[0121] ΔR=δΩ×R=S(δΩ)R (8)

[0122] Where: △R represents the rotation matrix error; δΩ represents the attitude deviation of the moving platform of the mechanism; S(δΩ) is the antisymmetric matrix of δΩ.

[0123] Solve for the antisymmetric matrix S(δΩ) in equation (8), and then solve for the vector δΩ using the properties of the antisymmetric matrix, and write it in the form of matrix multiplication:

[0124]

[0125] Among them: J wp Jacobian mapping matrix representing the deviation of the moving platform of the mechanism.

[0126] Substituting equations (7) and (9) into equation (6), we get:

[0127]

[0128] Where: S(Rd) represents the antisymmetric matrix of the product of Rd.

[0129] In equation (19), q i Moving (i = 1, 2, 3) to the right side of the equation and substituting the signs from the error modeling part, we get three equations (equ1, equ2, equ3). Then, performing total differential on both sides of the equations yields:

[0130]

[0131] Among them: A n (n = 1, 2, 3) represents the unknown, Bn (n = 1, 2, 3) represents the unknowns, C n (n = 1, 2, 3) represents the unknown, D n (n = 1, 2, 3) represents the unknown, E n (n = 1, 2, 3) represents the unknown, F n (n = 1, 2, 3) represents the unknown; △l an (n=1,2,3) represents the error of the circumcircle radius of the static plateau on branch n, Δl bn (n=1,2,3) represents the error of the circumcircle radius of the moving platform on branch n; △l n (n = 1, 2, 3) represents link A on branch n. n B n Length error.

[0132]

[0133] In the formula: Represents equ i For l ai Find the partial derivative; Represents equ i For l bi Find the partial derivative; Represents equ i For l i Find the partial derivative; Represents equ i Take the partial derivative with respect to α; Represents equ i Take the partial derivative with respect to β; Represents equ i Take the partial derivative with respect to z0; This represents integral operations.

[0134] Taking (Δα, Δβ, Δz0) as the unknown quantity, Δε 0 As the parameters of the equation, equation (11) is expressed in matrix form:

[0135]

[0136] in:

[0137]

[0138]

[0139] Δl a =[δl a1 δl a2 δl a3 ] T

[0140] Δlb =[δl b1 δl b2 δl b3 ] T

[0141] Δl=[δl1 δl2 δl3] T

[0142] In the formula: δl an (n = 1, 2, 3) represents the radius of the circumcircle of the static platform; δl bn (n = 1, 2, 3) represents the radius of the circumcircle of the moving platform; δl n (n = 1, 2, 3) represents the length of the branch link; Δl a This represents the error in the radius of the circumscribed circle of the static platform; Δl b This represents the error in the circumscribed circle radius of the moving platform; Δl n This represents the error in the length of the branch link.

[0143] This establishes the mapping relationship between the kinematic error of the mechanism and the end-effector attitude error of the moving platform:

[0144] [Δα Δβ Δz0] T =J e Δε 0 (14)

[0145] In the formula: J e The Jacobian matrix represents the relationship between the residual kinematic error of the mechanism and the end-effector attitude error of the moving platform; Δε 0 Represents the remaining kinematic error parameters and:

[0146]

[0147] Substituting equation (14) into equation (10) yields the final error mapping model for the 2PRU-PSR mechanism as follows:

[0148] Δr=JΔε (15)

[0149] In the formula: J represents the error mapping Jacobian matrix and the kinematic error parameters, specifically:

[0150] J = [J q R],Δε=[Δε 0 Δd] T

[0151] J q The matrix is ​​a partial Jacobian matrix expression in matrix form after simplification of equation (10); it is expressed as follows:

[0152] J q =J p J e-S(Rd)J wp J e (16)

[0153] In step three, the least squares sum of squares optimization method is selected for identification calculation; the attitude error and position error before and after kinematic calibration are compared to verify the correctness of the built model.

[0154] In step four, a BP (Back Propagation) neural network is selected to predict the remaining localization error. The network topology for this prediction is determined to be a BP neural network model consisting of an input layer, a hidden layer, and an output layer.

[0155] Before using the network for error prediction, the network model needs to be trained with a large amount of data. The network is configured with 6 inputs and 1 output, and the purelin function is selected as the network transfer function. Gradient training is used. The maximum number of training iterations is set to 5000, the learning rate is 0.01, the minimum allowable error is set to 0.0001, and the weights and thresholds are randomly set to 1 or 2.

[0156] Referring to the following empirical formula, different numbers of hidden layer nodes can be selected to train the 2PRU-PSR mechanism on data samples.

[0157]

[0158] In the formula, p represents the number of hidden layer nodes, m and n represent the number of input layer and output layer nodes, and a is an adjustment parameter ranging from 1 to 10.

[0159] Under the same training sample size and learning rate, different numbers of hidden layer nodes were selected for step-by-step experiments, and prediction simulations were performed after the network model was trained. Ultimately, it was determined that the number of hidden layer nodes was 10 for both training pose error data and training position error data.

[0160] Training was performed on different numbers of samples to find the best performance. Then, the 2PRU-PSR parallel robot was kinematically calibrated and the prediction error was compensated based on a neural network-based non-parametric model.

[0161] Example 1 (Error Compensation for 2PRU-PSR Mechanism Model)

[0162] like Figure 4 The diagram shown is a simplified representation of the mechanism of the 2PRU-PSR parallel robot, which consists of a moving platform, a stationary platform, and three branches.

[0163] The 2PRU-PSR is a 3-DOF parallel mechanism with two specific rotational axes and one translational axis (2R1T). U, P, S, and R represent Hooke joint, prismatic joint, spherical joint, and revolute joint, respectively. The coordinate systems for the static and moving platforms, i.e., the fixed coordinate system and the motion coordinate system (O-xyz and o-uvw), are established as shown in the figure, and C... n (n = 1, 2, 3) are all located on the static platform plane. A1 represents the connection point between the revolute and prismatic joints on branch 1, A2 represents the connection point between the revolute and prismatic joints on branch 2; A3 represents the connection point between the spherical and prismatic joints on branch 3; B1 represents the connection point between the Hooke joint and revolute joint on branch 1, B2 represents the connection point between the Hooke joint and revolute joint on branch 2, B3 represents the connection point between the Hooke joint and revolute joint on branch 3; C1 represents the projection point of the guide rail on branch 1 onto the fixed plane, C2 represents the projection point of the guide rail on branch 2 onto the fixed plane, and C3 represents the projection point of the guide rail on branch 4 onto the fixed plane. The motion coordinate system on the moving platform is o-uvw, with the origin o at the midpoint of B1B2; the u-axis points to oB3, the v-axis always points towards oB1, and the w-axis is perpendicular to the moving platform. For the fixed coordinate system O-xyz, the origin O is located at the center point of C1C2. The x-axis points to OC3, the y-axis always points towards OC1, and the z-axis direction can be obtained using the right-hand rule. This paper defines the system architecture parameters of the 2PRU-PSR mechanism as follows: the circumscribed circle radius of the static platform is OC1 = la1, OC2 = la2, OC3 = la3, and la1 = la2 = la3 = R; the link length is A1B1 = l1, A2B2 = l2, A3B3 = l3, and l1 = l2; the circumscribed circle radius of the moving platform is oB1 = lb1, oB2 = lb2, oB3 = lb3, and lb1 = lb2 = e1, lb3 = e2.

[0164] The direction of motion C of the three sliding joints n A n (n = 1, 2, 3) are parallel to each other. The moving platform of the mechanism is connected to the stationary platform through three branches: two PRU branches and one PSR branch. In the PRU branches, the axis of the R joint on links A1B1 and A2B2 is perpendicular to the link direction and parallel to the first rotation axis of the U joint. The second rotation axes of the U joints in the two PRU branches coincide to ensure the rotational output of the moving platform. In the PSR branch, the R joint at the end of link A3B3 is connected to the moving platform, and its rotation axis is parallel to the second rotation axis of the U joint in the PRU chain. The mechanism adjusts the position and orientation of the output moving platform by driving three moving joints P.

[0165] Step 1

[0166] The closed-loop vector equations for the three branches are as follows:

[0167] q n =p+bn -c n (n=1,2,3) (18)

[0168] Where: q n (n = 1, 2, 3) represents the driving value of the traverse joint on branch n, p represents the position vector from the origin O of the fixed coordinate system to the origin o of the moving coordinate system, and b n and c n These represent the points from the origin o of the moving coordinate system to B. n The position vector of the point oB n and by A n Point to B n The position vector A of the point n B n .

[0169] The rotation matrix from the moving coordinate system to the fixed coordinate system can be expressed as:

[0170]

[0171] Based on the mechanism characteristics of 2PRU-PSR, the origin of the moving coordinate system always moves on the plane x=0. Using the relationship between the vector p minus the vector OA3 (p-OA3) in the closed loop O-A3-B3-o and the perpendicularity of the joint axis vector at B3, the coordinate expression of the position vector p can be written as:

[0172] p=(0 (q3-z0)t α z0) T (20)

[0173] In formulas (19) and (20), △ O R o represents the rotation matrix error from the moving coordinate system to the fixed coordinate system; s, c, and t represent the abbreviations of the sine, cosine, and tangent function operators, respectively; α and β represent the output angles of the moving platform, and z0 represents the z-axis coordinate of the moving platform; s α c represents the sine value of α. α t represents the cosine value of α. α s represents the tangent of α; β c represents the sine value of β. β t represents the cosine value of β. β represents the tangent of β; T represents the inverse operation of the matrix.

[0174] Position vector b i The coordinate expression is:

[0175]

[0176] In the formula: the radius of the circumscribed circle of the moving platform oBn =lb n (n=1,2,3), and lb1=lb2=e1, lb3=e2.

[0177] The inverse kinematics expressions of the 2PRU-PSR mechanism are obtained by solving formulas (18)-(21), that is, the analytical expressions of the three moving pairs (q1, q2, q3) are as follows:

[0178]

[0179] In the formula: α and β represent the output angles of the moving platform, and z0 represents the z-axis coordinate of the moving platform; l n (n = 1, 2, 3) represents link A n B n The length of ; R represents the rotation matrix.

[0180] The position coordinate expression of the end effector of the mechanism can be solved using the following relationship:

[0181] r = p + O R o d (23)

[0182] In the formula: r represents the position of the end effector of the mechanism; p represents the position vector from the origin O of the fixed coordinate system to the origin o of the moving coordinate system.

[0183] Where vector d represents the coordinates of the end effector position in the moving platform coordinate system, that is:

[0184]

[0185] In the formula: r n (n = x, y, z) represent the components of r on the x, y, and z axes, respectively; d represents the coordinates of the end effector of the mechanism in the moving platform coordinate system; d n (n = x, y, z) represent the components of d on the x, y, and z axes, respectively.

[0186] Finally, the coordinates of the end effector position r obtained by measurement are used to inversely solve for the output angle and position information of the moving platform.

[0187] Substituting (α, β, z0) into equation (22) yields the three prismatic drive values ​​required to move the end effector of the mechanism to the specified position. Thus, the inverse kinematic model of the 2PRU-PSR parallel mechanism is obtained:

[0188] q = [q1 q2 q3] T =g(r,ε+Δε) (25)

[0189] Where: q represents the total driving value of the mechanism; qn (n=1,2,3) represents the driving value of the moving pair on branch n; T represents the inverse solution operation of the matrix; ε represents the nominal kinematic parameters of the mechanism; Δε represents the kinematic error of the mechanism; g(r,ε+Δε) represents the functional relationship with r and ε+Δε as parameters.

[0190] The forward kinematics section uses Newton's iteration method to derive the output platform end-effector pose information (α, β, z0) and the mechanism end-effector position information (r) corresponding to the driving inputs (q1, q2, q3). x r y r z ).

[0191] Newton's iterative formula is as follows:

[0192]

[0193] In the formula: x n x represents the nth iteration. n+1 f'(x) represents the (n+1)th iteration; n ) is f(x) n The derivative of ).

[0194] First, the inverse solution of equation (22) is written in matrix form as follows:

[0195]

[0196] In the formula: Q represents a matrix.

[0197] After successively differentiating the moving platform pose information (α, β, z0) in the inverse solution matrix to obtain the derivative matrix, the derivative matrix is ​​substituted into Newton's iteration formula to set the initial values ​​of the iteration and the precision condition for the loop termination. The final moving platform end-effector pose information (α, β, z0) corresponding to each drive group is obtained through iterative updates. After calculating the pose parameters, the rotation matrix... O R o It can also be determined that the position r of the end effector of the mechanism is calculated by equation (23), thus obtaining the complete forward kinematic model of the 2PRU-PSR parallel mechanism with the end effector:

[0198] r = [r x r y r z ] T =f(q,ε+Δε) (28)

[0199] In the formula: f(q,ε+Δε) represents the functional relationship with r and ε+Δε as parameters.

[0200] Step Two

[0201] In the process of modeling the error mapping model of the 2PRU-PSR mechanism, a total of 12 geometric error parameters were determined, and their corresponding structural system parameters in the mechanism are shown in the table below:

[0202] Table 1. Kinematic error item number and explanation

[0203]

[0204] First, by applying a linear perturbation to both sides of equation (23), we can obtain:

[0205] Δr=Δp+Δ O R o d+ O R o Δd=JΔε (29)

[0206] △r represents the position error of the end effector of the mechanism; △p represents the position vector error from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; △ O R O Δd represents the rotation matrix error from the moving coordinate system to the fixed coordinate system; Δd represents the coordinate error of the end effector position in the moving platform coordinate system; J represents the error mapping Jacobian matrix and kinematic error parameters.

[0207] By substituting the symbols from the error modeling part back into equation (20) and simplifying, we get:

[0208] p = [0 y0 z0] T (30)

[0209] in

[0210] In the formula: l an (n = 1, 2, 3) represents the radius of the circumcircle of the static plateau on branch n, l bn (n = 1, 2, 3) represents the radius of the circumcircle of the moving platform on branch n; l n (n = 1, 2, 3) represents link A on branch n. n B n The length.

[0211] Linear perturbation of equation (30) and its matrix multiplication form is as follows:

[0212]

[0213] Among them, J pThe matrix represents the Jacobian matrix of the position vector p after linear perturbation with respect to the attitude error of the moving platform's end effector; △α and △β represent the output angle error of the moving platform, and △z0 represents the z-axis coordinate error of the moving platform; and J p1 J p2 They are respectively:

[0214]

[0215]

[0216] According to the differential formula for the rotation matrix, the rotation matrix ΔR can be expressed as:

[0217] ΔR=δΩ×R=S(δΩ)R (32)

[0218] Where: △R represents the rotation matrix error; δΩ represents the attitude deviation of the moving platform of the mechanism; and S(δΩ) represents the antisymmetric matrix of δΩ.

[0219] Solve for the antisymmetric matrix S(δΩ) in equation (32), and then solve for the vector δΩ using the properties of the antisymmetric matrix, and write it in the form of matrix multiplication:

[0220]

[0221] Among them: J wp Let be the Jacobian mapping matrix of the deviation of the moving platform of the mechanism.

[0222] Substituting equations (31) and (33) into equation (29), we get:

[0223]

[0224] Where: S(Rd) represents the antisymmetric matrix of the product of Rd.

[0225] Next, the end-effector attitude error (Δα, Δβ, Δz0) and the residual kinematic error parameter Δε of the moving platform are calculated. 0 The mapping relationship between them. Where: Δε 0 =[Δl a Δl b Δl] T .

[0226] In equation (22), q i Moving (i = 1, 2, 3) to the right side of the equation and substituting the signs from the error modeling part, we get three equations (equ1, equ2, equ3). Then, performing total differential on both sides of the equations yields:

[0227]

[0228] Among them: An (n = 1, 2, 3) represents the unknown, B n (n = 1, 2, 3) represents the unknowns, C n (n = 1, 2, 3) represents the unknown, D n (n = 1, 2, 3) represents the unknown, E n (n = 1, 2, 3) represents the unknown, F n (n = 1, 2, 3) represents the unknown; △l an (n=1,2,3) represents the error of the circumcircle radius of the static plateau on branch n, Δl bn (n=1,2,3) represents the error of the circumcircle radius of the moving platform on branch n; △l n (n = 1, 2, 3) represents link A on branch n. n B n Length error.

[0229]

[0230] In the formula: Represents equ i For l ai Find the partial derivative; Represents equ i For l bi Find the partial derivative; Represents equ i For l i Find the partial derivative; Represents equ i Take the partial derivative with respect to α; Represents equ i Take the partial derivative with respect to β; Represents equ i Take the partial derivative with respect to z0; This represents integral operations.

[0231] At this point, with (Δα, Δβ, Δz0) as the unknowns, Δε 0 As the parameters of the equation, equation (35) is expressed in matrix form:

[0232]

[0233] in:

[0234]

[0235]

[0236] Δl a =[δl a1 δl a2 δl a3 ] T

[0237] Δl b =[δl b1 δl b2 δl b3 ] T

[0238] Δl=[δl1 δl2 δl3] T

[0239] In the formula: δl an (n = 1, 2, 3) represents the radius of the circumcircle of the static platform; δl bn (n = 1, 2, 3) represents the radius of the circumcircle of the moving platform; δl n (n = 1, 2, 3) represents the length of the branch link; Δl a This represents the error in the radius of the circumscribed circle of the static platform; Δl b This represents the error in the circumscribed circle radius of the moving platform; Δl n This represents the error in the length of the branch link.

[0240] This establishes the mapping relationship between the kinematic error of the mechanism and the end-effector attitude error of the moving platform:

[0241] [Δα Δβ Δz0] T =J e Δε 0 (37)

[0242] Among them, J e The Jacobian matrix represents the relationship between the residual kinematic error of the mechanism and the end-effector attitude error of the moving platform; Δε 0 Represents the remaining kinematic error parameters and:

[0243]

[0244] Substituting equation (37) into equation (34) yields the final error mapping model for the 2PRU-PSR mechanism as follows:

[0245] Δr=JΔε (38)

[0246] Where J represents the error mapping Jacobian matrix and the kinematic error parameters, specifically:

[0247] J = [J q R],Δε=[Δε 0 Δd] T

[0248] J q The matrix is ​​a partial Jacobian matrix expression in matrix form after simplification of equation (34), as follows:

[0249] J q=J p J e -S(Rd)J wp J e (39)

[0250] Step 3

[0251] After establishing the inverse kinematics, forward kinematics, and error models, calibrating and simulating the mechanism using MATLAB can effectively verify the correctness of the established models and the accuracy of parameter identification.

[0252] The parameter errors were set, and points were randomly selected within the workspace. For the 12 error parameters given in the error model, errors were randomly generated between 0 and 1. Theoretically, the final calibration result is not affected by the calibration points; therefore, 56 calibration points were randomly selected within the workspace of the 2PRU-PSR mechanism. Each calibration point contains two rotational degrees of freedom and one translational degree of freedom, representing motion in three degrees of freedom: α, β, and z0. After completing the pre-set steps, the selected calibration points were substituted into the inverse kinematics model without error parameters to calculate the theoretical driving values. These theoretical driving values ​​were then substituted into both the forward kinematics model without error parameters and the forward kinematics model with error parameters to obtain theoretical pose information and actual pose information representing the measured values ​​in the calibration experiment. Since a neural network is needed to predict and compensate for the remaining unidentifiable geometric and non-geometric errors after calibration, and the neural network model has a weak ability to identify errors, and the data itself undergoes noise reduction and normalization before being input into the network structure, measurement noise was not added to the simulation.

[0253] Then, the error mapping matrix calculated in the error model, i.e., the identification matrix, is used to select an appropriate identification algorithm for parameter identification. Since the parameter identification of parallel mechanisms can usually be regarded as an optimization problem of minimizing the objective function, this paper chooses to use the least squares sum of squares optimization method for identification calculation.

[0254] After multiple iterations of identification, the identified error parameters can be obtained. These error parameters are then compensated into the kinematic parameters, and the forward kinematics solution with the identified error parameters is calculated to obtain the identified and optimized pose information of the moving platform. A comparison of attitude and position errors before and after kinematic calibration is provided. Figure 5 As shown.

[0255] In Figure (a), the average L2 norm of the attitude error vector before calibration decreased from 0.0176 rad to 0.00187 rad. In Figure (b), the average L2 norm of the position error vector before calibration decreased from 2.72 mm to 0.0781 mm. The calibration effect is significant, verifying the correctness of the kinematic and error models, but some residual errors still exist. Assuming that the residual errors are caused by some unidentifiable geometric parameter errors and non-geometric parameter errors, we will further optimize and compensate for the residual errors using a non-parametric model based on neural networks.

[0256] Step Four

[0257] A back propagation (BP) neural network was chosen to predict the remaining localization error. The network topology for this prediction was determined to be a BP neural network model consisting of one input layer, one hidden layer, and one output layer.

[0258] Before using a network for error prediction, the network model needs to be trained with a large amount of data. First, the input and output of the network model are determined. Based on the input and output data samples, the number of nodes in each layer of the network structure is reasonably set, and the initial values ​​of the connection weights of each layer and the output thresholds of the hidden layers are set. Network parameters such as the number of training iterations and the learning rate are also determined. After importing the training data, it is necessary to normalize the data. The purelin function is selected as the network transfer function, and gradient training is used. The maximum number of training iterations is set to 5000, the learning rate is 0.01, the minimum allowable error is set to 0.0001, and the weights and thresholds are randomly set to 1 or 2.

[0259] The input to the network structure is determined as the identified end-effector pose information after kinematic calibration, and the output is determined as the residual error after calibration.

[0260] The remaining pose information error of the robot after calibration is selected, namely the pose error vectors of the three moving platforms (Δα, Δβ, Δz0) and the position error vectors of the three end effectors (Δr). x , Δr y , Δr z However, after actual training, it was found that if the network is trained with 6 inputs and 6 outputs, the number of feature terms in the input sample data structure is small, while the number of categories in the output sample data structure is too large. This results in poor prediction performance after network training, and it can only predict errors that are very similar to the data structures of the training samples and the prediction samples. Therefore, the network output is further modified to be the vector sum of the pose error and the position error as shown in Equation (38), which predicts the remaining pose error and position error respectively. After prediction using the parameterless model, secondary compensation is performed to directly compensate for the remaining error.

[0261]

[0262] Therefore, the network is configured with 6 inputs and 1 output. Referring to the following empirical formula, different numbers of hidden layer nodes are selected to train the 2PRU-PSR mechanism on data samples.

[0263]

[0264] Where p represents the number of hidden layer nodes, m and n represent the number of input and output layer nodes, and a is an adjustable parameter ranging from 1 to 10. Under the same training sample size and learning rate conditions, different numbers of hidden layer nodes were selected for step-by-step experiments. After training the network model, prediction simulations were performed, and finally, the predicted values ​​of the network based on 21 sets of test samples were compared with the actual output values ​​of the samples. It was ultimately determined that the number of hidden layer nodes should be 10 for both training pose error data and training position error data.

[0265] The optimal training data sample size was determined by gradually increasing the number of calibration points from 112 to approximately 1008. The table below shows the post-training prediction simulation results with training data samples of 112, 560, and 1008 sets respectively:

[0266] Table 2. Absolute average of prediction simulation error under different training sample sizes.

[0267]

[0268] As shown in the table above, the neural network's predictive ability is best when trained with 1008 sets of samples, meeting the requirement of predicting the remaining unidentifiable geometric and non-geometric errors.

[0269] The identified end-effector pose information corresponding to the 56 kinematically calibrated points mentioned above is used as input samples to the network model for prediction. It is important to note that the training and testing samples for the neural network must not overlap with the data from these calibration points. The remaining pose and position errors are predicted separately, and secondary compensation is performed after prediction using the non-parametric model, directly compensating for the remaining errors. The comparison of the prediction error compensation before and after using the non-parametric model based on the neural network is shown below. Figure 6 As shown.

[0270] After prediction and compensation by a BP neural network, Figure 6 In case a, the average L2 norm of the attitude error vector decreased from 0.00187 rad before prediction to 0.00101 rad. Figure 6The average L2 norm of the position error vector in b decreased from 0.0781 mm before prediction to 0.0496 mm. This shows that the prediction compensation significantly reduced the error, demonstrating the feasibility of the neural network-based non-parametric model prediction error compensation method and its ability to compensate for the shortcomings of kinematic calibration. This resulted in a second improvement in the positioning accuracy of the 2PRU-PSR parallel robot.

Claims

1. An error compensation method for a 2PRU-PSR parallel robot, comprising the following steps: Step 1: Establish the inverse kinematics model of the 2PRU-PSR parallel robot using the closed-loop vector method and calculate the complete forward kinematics model using Newton's iterative formula; Step 2: Set the error parameters and construct the 2PRU-PSR parallel robot error mapping model using the vector differential method as follows: The position error of the end effector of the mechanism is represented by J; J represents the error mapping Jacobian matrix and kinematic error parameters; Δε represents the kinematic error of the mechanism. Step 3: Complete the identification of kinematic error parameters and perform calibration simulation to verify the correctness of the established model; Step 4: Use a non-parametric model based on neural networks to predict the residual error at the end of the 2PRU-PSR parallel robot after kinematic calibration, and use the non-parametric model to predict the error to complete the compensation. In step two, the error mapping model construction process for the 2PRU-PSR parallel robot is as follows: The position coordinates of the end effector r can be expressed as: （4） In the formula: p represents the position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; O R o d represents the rotation matrix from the moving coordinate system to the fixed coordinate system; d represents the coordinates of the end effector position of the mechanism in the moving platform coordinate system. Applying linear perturbation to both sides of equation (4) yields: （10） In the formula: S(Rd) represents the antisymmetric matrix of the product of Rd; R represents the rotation matrix; Δd represents the coordinate error of the end effector position in the moving platform coordinate system; J p The matrix represents the Jacobian matrix of the position vector p with respect to the attitude error at the end of the moving platform after linear perturbation; J wp The Jacobian mapping matrix represents the deviation of the moving platform of the mechanism; △α and △β represent the output angle error of the moving platform, and △z0 represents the z-axis coordinate error of the moving platform. Taking (Δα, Δβ, Δz0) as the unknown quantity, Δε 0 Using the equation parameters, we obtain the mapping relationship between the mechanism's kinematic error and the end-effector attitude error of the moving platform: （14） In the formula: J e The Jacobian matrix represents the relationship between the residual kinematic error of the mechanism and the end-effector attitude error of the moving platform; Δε 0 Represents the remaining kinematic error parameters and: Substituting equation (14) into equation (10) yields the final error mapping model for the 2PRU-PSR mechanism as follows: （15） In the formula, J represents the error mapping Jacobian matrix and the kinematic error parameters, specifically: J q The matrix is ​​a partial Jacobian matrix expression in matrix form after simplification of equation (10), as follows: （16）。 2. The 2PRU-PSR parallel robot error compensation method according to claim 1, characterized in that: In step one, the inverse kinematic model of the 2PRU-PSR parallel mechanism is expressed as: （1） In the formula, q represents the total driving value of the mechanism; n (n=1,2,3) represents the driving value of the moving pair on branch n; T represents the inverse solution operation of the matrix; g represents the functional relationship; r represents the position of the end effector of the mechanism; ε represents the nominal kinematic parameters of the mechanism; Δε represents the kinematic error of the mechanism; g(r, ε+Δε) represents the functional relationship with r and ε+Δε as parameters; The inverse kinematics expressions for the three moving secondary drive values ​​q1, q2, and q3 are shown below: （19） In the formula: α and β represent the output angles of the moving platform, z0 represents the z-axis coordinate of the moving platform, and the radius of the circumscribed circle of the moving platform is oB. n =lb n (n=1,2,3), and lb1=lb2=e1, lb3=e2; s, c, and t represent the abbreviations of the sine, cosine, and tangent functions, respectively; s α c represents the sine value of α. α t represents the cosine value of α. α s represents the tangent of α; β c represents the sine value of β. β t represents the cosine value of β. β The value representing the tangent of β; l n (n=1,2,3) represents link A n B n The length of ; R represents the rotation matrix.

3. The 2PRU-PSR parallel robot error compensation method according to claim 2, characterized in that: Newton's iteration method is chosen to obtain the end-effector pose information and end-effector position information corresponding to the driving input. The Newton's iteration formula is expressed as: （2） In the formula: x n x represents the nth iteration. n+1 This represents the (n+1)th iteration; f'( x n ) is f(x) n The derivative of ) The inverse kinematics expression of the moving pair drive is written in matrix form. Then, the derivative matrix is ​​obtained by successively differentiating the pose information (α, β, z0) of the moving platform in the inverse matrix. Substituting these derivative matrices into Newton's iteration formula to set the initial values ​​of the iteration and the accuracy conditions for the termination of the cycle, a complete forward kinematics model of the 2PRU-PSR parallel mechanism with an end effector is obtained. （3） In the formula: r represents the position of the end effector of the mechanism; r n (n=x, y, z) represent the components of r on the x, y, and z axes, respectively; f(q, ε+Δε) represents the functional relationship with parameters r and ε+Δε.

4. The 2PRU-PSR parallel robot error compensation method according to claim 3, characterized in that: In step two, the error mapping model construction process for the 2PRU-PSR parallel robot is as follows: 1) The position coordinates of the end effector r can be expressed as: （4） In the formula: p represents the position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; O R o d represents the rotation matrix from the moving coordinate system to the fixed coordinate system; d represents the coordinates of the end effector position of the mechanism in the moving platform coordinate system. The position vector pointing from the origin O of the fixed coordinate system to the origin o of the moving coordinate system can be represented as: （5） Where y0 represents the y-axis coordinate of the moving platform, specifically: In the formula: l an (n=1,2,3) represents the radius of the circumcircle of the static plateau on branch n, l bn (n=1,2,3) represents the radius of the circumcircle of the moving platform on branch n; l n (n=1,2,3) represents link A n B n Length; 2) Applying linear perturbation to both sides of equation (4) yields: （6） △r represents the position error of the end effector of the mechanism; △p represents the position vector error from the origin O of the fixed coordinate system to the origin o of the moving coordinate system; △ O R o Δd represents the rotation matrix error from the moving coordinate system to the fixed coordinate system; Δd represents the coordinate error of the end effector position in the moving platform coordinate system; J represents the error mapping Jacobian matrix and kinematic error parameters. Linear perturbation of equation (5) and its matrix multiplication form is as follows: （7） Among them: J p The matrix represents the Jacobian matrix of the position vector p after linear perturbation with respect to the attitude error of the moving platform's end effector; △α and △β represent the output angle error of the moving platform, and △z0 represents the z-axis coordinate error of the moving platform; and J p1 J p2 They are respectively: According to the differential formula of the rotation matrix, ΔR can be expressed as: （8） Where: △R represents the rotation matrix error; δΩ represents the attitude deviation of the moving platform of the mechanism; S(δΩ) is the antisymmetric matrix of δΩ; Solve for the antisymmetric matrix S(δΩ) in equation (8), and then solve for the vector δΩ using the properties of the antisymmetric matrix, and write it in the form of matrix multiplication: （9） Among them: J wp Jacobian mapping matrix representing the deviation of the moving platform of the mechanism; 3) Substituting equations (7) and (9) into equation (6), we get: （10） Where: S(Rd) represents the antisymmetric matrix of the product of Rd; 4) In equation (19), q i Moving (i=1, 2, 3) to the right side of the equation and substituting the signs from the error modeling part, we get three equations (equ1, equ2, equ3). Then, performing total differential on both sides of the equations yields: （11） Among them: A n (n=1,2,3) represents the unknowns, B n (n=1,2,3) represents the unknowns, C n (n=1,2,3) represents the unknowns, D n (n=1,2,3) represents the unknown, E n (n=1,2,3) represents the unknowns, F n (n=1,2,3) represents the unknown; △l an (n=1,2,3) represents the error of the circumcircle radius of the static plateau on branch n, Δl bn (n=1,2,3) represents the error of the circumscribed circle radius of the moving platform on branch n; △l n (n=1,2,3) represents link A on branch n. n B n Length error; （12） In the formula: (i=1,2,3) represents equation i For l ai Find the partial derivative; (i=1,2,3) represents equation i For l bi Find the partial derivative; (i=1,2,3) represents equation i For l i Find the partial derivative; (i=1,2,3) represents equation i Take the partial derivative with respect to α; (i=1,2,3) represents equation i Take the partial derivative with respect to β; (i=1,2,3) represents equation i Take the partial derivative with respect to z0; ∂ represents integration. Taking (Δα, Δβ, Δz0) as the unknown quantity, Δε 0 As the parameters of the equation, equation (11) is expressed in matrix form: （13） in: In the formula: δl an (n=1,2,3) represents the radius of the circumcircle of the static platform; δl bn (n=1,2,3) represents the radius of the circumcircle of the moving platform; δl n (n=1,2,3) represents the length Δl of the branch link. a This represents the error in the radius of the circumscribed circle of the static platform; Δl b This represents the error in the circumscribed circle radius of the moving platform; Δl n This represents the error in the length of the branch link; This establishes the mapping relationship between the kinematic error of the mechanism and the end-effector attitude error of the moving platform: （14） In the formula: J e The Jacobian matrix represents the relationship between the residual kinematic error of the mechanism and the end-effector attitude error of the moving platform; Δε 0 Represents the remaining kinematic error parameters and: 5) Substituting equation (14) into equation (10) yields the final error mapping model of the 2PRU-PSR mechanism as follows: （15） In the formula, J represents the error mapping Jacobian matrix and the kinematic error parameters, specifically: J q The matrix is ​​a partial Jacobian matrix expression in matrix form after simplification of equation (10), as follows: （16）。 5. The 2PRU-PSR parallel robot error compensation method according to claim 4, characterized in that: In step three, the least squares sum of squares optimization method is selected to identify and calculate the kinematic error parameters; the attitude error and position error before and after kinematic calibration are compared to verify the correctness of the built model.

6. The 2PRU-PSR parallel robot error compensation method according to claim 5, characterized in that: In step four, a BP (Back Propagation) neural network is selected to predict the remaining localization error. The predicted network topology is determined to be a BP neural network model consisting of an input layer, a hidden layer, and an output layer. Before using a network for error prediction, a large amount of data is needed to train the network model.

7. The 2PRU-PSR parallel robot error compensation method according to claim 6, characterized in that: The network model being trained is configured with 6 inputs and 1 output, and the purelin function is selected as the network transfer function. Gradient training is used for training. The maximum number of training iterations is set to 5000, the learning rate is 0.01, the minimum allowable error is set to 0.0001, and the weights and thresholds are randomly set to 1 or 2. Referring to the following empirical formula, different numbers of hidden layer nodes can be selected to train the 2PRU-PSR mechanism on data samples; （17） In the formula, p represents the number of hidden layer nodes, m and n represent the number of input layer and output layer nodes, and a is an adjustment parameter ranging from 1 to 10; Under the same training sample size and learning rate, different numbers of hidden layer nodes were selected for step-by-step experiments, and prediction simulations were performed after the network model was trained. Training was performed on different numbers of samples to find the best performance. Then, the 2PRU-PSR parallel robot was kinematically calibrated and the prediction error was compensated based on a neural network-based non-parametric model.

8. The 2PRU-PSR parallel robot error compensation method according to claim 7, characterized in that: The number of nodes in each hidden layer is 10.

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