Parallel processing robot kinematics calibration method considering passive joint error

Abstract

The invention relates to the technical field of robot calibration, and aims to provide a parallel processing robot kinematics calibration method considering passive joint errors so as to improve the absolute positioning precision of an end effector of a parallel robot and further improve the application efficiency of the parallel robot in the manufacturing industry. According to the technical scheme, the method comprises the following steps: step 1, constructing a sphere representing the motion space of each branch end point under a conformal geometric algebra framework; 2, a kinematical equation of the parallel machining robot is obtained through operation, and then perturbation is conducted to obtain an initial robot error model; 3, obtaining a complete robot error model containing passive joint errors; 4, the actual pose of the robot is obtained, and actual pose data are input into the complete robot error model; and step 5, taking the deviation between the actual pose and the nominal pose of the robot as an optimization target, identifying error parameters in the complete robot error model by adopting a regularization Levenberg-Marquardt algorithm, and finally compensating kinematics parameters.

Description

Technical Field

[0001] The present invention relates to the technical field of robot calibration, and in particular to a kinematic calibration method for a parallel machining robot taking into account passive joint errors. Background Art

[0002] Parallel robots offer advantages such as high rigidity and good load-bearing capacity, making them suitable for the efficient machining of large aerospace structures. Kinematic calibration is an effective means of ensuring the accuracy of parallel robots and improving machining quality. It involves four key steps: error modeling, error measurement, parameter identification, and error compensation. This approach aims to improve the absolute positioning accuracy of the parallel robot's end effector.

[0003] Accurate error modeling is an important prerequisite for ensuring the effectiveness of kinematic calibration, but there are still problems with the passive joint errors of parallel robots, such as unclear physical meaning, inability to directly measure, and poor identifiability. To ensure the accuracy of parallel robots, two error modeling techniques are commonly used in kinematic calibration: one is to establish the inverse kinematic equation and then fully differentiate it to obtain the robot's error model. However, this method has some ideal assumptions and is difficult to meet the integrity requirement of including all error parameters in the error model. It also does not include passive joint errors, which limits the improvement of accuracy. The other is to establish the error model of the entire robot based on the error transmission model of the series branch chain. However, this method also cannot directly identify the errors of the passive joints, which brings difficulties to the precise control and error compensation of the mechanism.

[0004] At present, there is a wealth of research on the kinematic calibration of parallel robots. For example, Chinese patent CN118578383A (application number: 202410713317.3) analyzes the motion helices, constraint helices, and force transmission helices of each branch of the parallel module to construct an overall stiffness and flexibility model to complete the kinematic calibration of the five-axis parallel robot. Chinese patent CN113580148A (application number: 202111070626.6) is based on the equivalent kinematic chain method to construct a virtual serial motion branch of the parallel robot to obtain the pose matrix, perform full differential calculations on it to determine the error model of the parallel robot, and complete the kinematic calibration of the parallel robot.

[0005] Due to the presence of passive joint errors in parallel robots, it is difficult to develop a kinematic error model that meets integrity requirements. Existing kinematic calibration methods for parallel robots often only consider geometric errors, ignoring the impact of passive joint errors on accuracy. Therefore, it is essential to propose a new error modeling method that considers passive joint errors and, based on this, to develop a complete set of kinematic calibration methods applicable to any parallel robot. Summary of the Invention

[0006] The purpose of the present invention is to overcome the shortcomings of the above-mentioned background technology and provide a kinematic calibration method for a parallel machining robot taking into account passive joint errors, so as to improve the absolute positioning accuracy of the end effector of the parallel robot and thereby improve its application efficiency in the manufacturing industry.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] A kinematic calibration method for a parallel machining robot considering passive joint errors is performed as follows:

[0009] Step 1: Based on the geometric structure of the parallel machining robot, a sphere representing the motion space of each branch end point is constructed within the framework of conformal geometric algebra;

[0010] Step 2: The kinematic equations of the parallel machining robot are obtained by performing the intersection operation based on the sphere geometry, and then the initial robot error model is obtained by performing the perturbation operation.

[0011] Step 3: Express the robot's passive joint errors as a function of kinematic error parameters and driving quantities. Combined with the initial robot error model, a complete robot error model including passive joint errors is obtained.

[0012] Step 4: Use a laser tracker to measure the error of the end position of the parallel machining robot to obtain the actual position of the robot and input the actual position data into the complete robot error model;

[0013] Step 5: Taking the deviation between the actual and nominal robot poses as the optimization target, the regularized Levenberg-Marquardt algorithm is used to identify the error parameters in the complete robot error model, and finally the kinematic parameters are compensated.

[0014] The spheres in the motion space of the end points of each branch described in step 1 are as follows:

[0015] In robot analysis based on conformal geometry algebra, there are five orthogonal basis vectors in the conformal geometry space, including three Euclidean orthogonal basis vectors {e1, e2, e3} and two additional orthogonal product vectors {e + ,e -}, where the two zero vectors e0 and e ∞ represents the origin and the point at infinity.

[0016] For the parallel processing robot, the connection point between branch i (i = 1 to 3) and the moving platform is defined as Bi (i = 1 to 3), and point B3 is obtained by the outer product of the three spheres. Moving platform B i The distance between (i=1~3) is defined as the geometric parameter lb ij (i=1~3,j=13,i≠j);

[0017] With B1 as the center of the ball, lb 13 A sphere with radius S b1b3 Expressed as:

[0018]

[0019] With B2 as the center of the ball, lb 23 A sphere with radius is represented by:

[0020]

[0021] The connection point between branch i (i=1-3) and the driving pair is defined as Ai (i=1-3). The sphere with A3 as the center and l3 as the radius is expressed as:

[0022]

[0023] The steps for establishing the initial robot error model in step 2 are:

[0024] (1) Point B3 is at the intersection of three spheres. Expressed as:

[0025]

[0026] Similarly, we can get the unique solution of point B3, which is expressed as follows:

[0027]

[0028] (2) The position of the robot's end effector is obtained using the vector method. The u-axis direction of the moving coordinate system is represented by the modulus of 2B3-B1-B2, the v-axis direction is represented by the modulus of B2-B1, the w-axis is the cross product of u and v, and k i (i=1-3) represents the movement of the end effector from point B2 along u, v and w respectively. That is, the end effector position p is expressed as:

[0029] p=B2+k1u+k2v+k3w (6)

[0030] From this, the end position p of the mechanism can be expressed as a function of the position equation F:

[0031] F(p,r m ,θ)=0 (7)

[0032] where r m Structural parameters of the parallel mechanism, θ represents the passive joint angle.

[0033] (3) By taking the total differential of the position equation of point B1 in the forward kinematic analysis, the mapping relationship between the position error of point B1 and the structural parameter error is obtained, which is expressed as follows:

[0034]

[0035] in They represent the differentials of the position equation of point B1 with respect to its position, structural parameters, and passive joint rotation angle; δr m represents the structural parameter error; δp b1 represents the position error of point B1; δθ represents the rotation angle error of the passive joint.

[0036] Formula (8) is organized into a matrix form and expressed as:

[0037]

[0038] in

[0039] Similarly, by taking the total differential of the position equations of points B1 and B2, we can obtain the error mapping relationship as follows:

[0040]

[0041] in δp b2 Indicates the position error of point B2, They represent the differentials of the position equation of point B2 with respect to its position, structural parameters, and passive joint rotation angle respectively;

[0042] δp b3 Indicates the position error of point B3, They represent the differentials of the position equation of point B3 with respect to its position, structural parameters, and passive joint rotation angle respectively;

[0043] (4) By differentiating the mechanism kinematic equation (6), the initial robot error model is obtained as follows:

[0044]

[0045] in Respectively represent the end position equation of the mechanism with respect to its own position and the moving platform B i Differentiation of point position and actuator structural parameters;

[0046] δp bi =[δx bi ;δy bi ;δz bi ](i=1~3) represents B i Point position error; δk i =[δk1; δk2; δk3] represents the actuator structural parameter error.

[0047] Formula Bi Substitute the point position error into equation (8) and write it in matrix form as follows:

[0048]

[0049] in Express the position equation F with respect to the actuator structural parameter k i The differential of Represents the structural parameter r k The actuator structural parameter error k i composition,

[0050] The initial robot error model without passive joint errors obtained by rearranging Equation (12) is expressed as:

[0051]

[0052] Where: δp is the end effector position error, represents the partial derivative of the position equation F with respect to the end effector position p, Represents the position equation F versus the structural parameter r m The partial derivative of δr m is the structural parameter error, represents the partial derivative of the position equation F with respect to the passive joint angle θ, and δθ is the passive joint angle error.

[0053] The process of establishing the complete robot error model in step 3 is as follows:

[0054] According to the geometric constraints of the robot, define the geometric angle between the link B1B2 and the link A3B3 is an invariant. Therefore, a single variable input-output equation about the passive joint angle θ is established, which is expressed as follows:

[0055]

[0056] Where l3 is the length of the connecting rod A3B3 of the mechanism. Since the motion error of the passive joint is caused by the error in the dimensional parameters of the mechanism, the motion error of the passive joint is mapped to the error in the dimensional parameters of the mechanism through the geometric relationship of the mechanism. By taking the total differential of Equation (14), the passive joint error is mapped into a function form of the kinematic error parameter and the driving amount, which is expressed as follows:

[0057]

[0058] in is the structural parameter error after mapping, is the error mapping matrix including the passive joint error. Combined with formula (13), the complete robot error model including the passive joint error is obtained as follows:

[0059]

[0060] where δp=[δx p ;δy p ;δz p ] represents the position error of the end effector; δr + Represents the structural parameter error after mapping; is the error mapping matrix containing the passive joint errors.

[0061] The specific steps of step 4 are:

[0062] T-Mac is used to measure the actual pose of the parallel mechanism end effector in the measurement coordinate system. The actual pose of the end effector in the mechanism coordinate system is obtained by combining the coordinate system transformation matrix, and the measured actual pose is used as the input of the error model.

[0063] Considering the current structural parameters In the case of the mechanism forward kinematics solution, the theoretical position p of the end effector is obtained (k) , the end pose p measured by the laser tracker + Combined with the error model, the error identification matrix under the current structural parameters is obtained

[0064] The derivation process of the structural parameter iterative identification formula used for identification in step 5 is:

[0065] Will and δp stacked into and δp + , and then solve the identification model Error of structural parameters δr + The optimal solution, that is, to find the appropriate δr + Let Minimum.

[0066] Because there may be In the case of insufficient rank, the use of nonlinear least squares fitting will lead to divergence in the identification results. Regularized nonlinear least squares methods can effectively prevent divergence during the identification process. Therefore, with the deviation between the actual position and the nominal position of the robot as the optimization target, the regularized Levenberg-Marquardt algorithm is selected to iteratively identify the error parameters. The formula for iterative identification of structural parameters based on the regularized Levenberg-Marquardt algorithm can be expressed as:

[0067]

[0068] Where λ represents the regularization coefficient. The iterative process is repeated until the robot error converges to improve the accuracy of the robot's terminal output and complete the kinematic calibration.

[0069] Compared with the existing calibration technology, the beneficial effects of the present invention are:

[0070] (1) The present invention aims to solve the problem of neglecting the passive joint error in the traditional calibration method, and to improve the absolute positioning accuracy of the end effector of the parallel robot by comprehensively considering the passive joint error factors, thereby improving its application efficiency in the manufacturing industry.

[0071] (2) This paper proposes a kinematic calibration method based on conformal geometric algebra for parallel robots. The forward kinematic analysis of the parallel robot is performed using the principle of geometric intersection. Based on this, a kinematic error model that meets the requirements of integrity and continuity is established. The error modeling process is highly geometrically intuitive, improving the absolute positioning accuracy of the robot's end-position. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 Flowchart of the kinematic calibration method of the present invention.

[0073] Figure 2 This is a schematic diagram of the 2-PRU-PUR parallel robot mechanism of Example 1 of the present invention.

[0074] Figure 3 This is a model diagram of the 2-PRU-PUR parallel robot of Example 1 of the present invention.

[0075] Figure 4 This is a schematic diagram of spatial geometric intersection in Example 1 of the present invention.

[0076] Figure 5 Diagram of the setup for the parallel robot kinematic calibration method considering passive joint errors.

[0077] Figure 6 This is the pose error diagram before and after the experimental calibration of Example 1 of the present invention. DETAILED DESCRIPTION

[0078] The principle and specific implementation methods of the present invention are further described below with reference to the accompanying drawings.

[0079] In order to solve the problems of unclear physical meaning, inability to directly measure, and poor identifiability of the passive joint errors of parallel robots, the present invention provides a kinematic calibration method for parallel machining robots considering passive joint errors, specifically (see the process for details). Figure 1 ):

[0080] Step 1: Based on the geometric structure of the parallel machining robot, a sphere representing the motion space of each branch end point is constructed within the framework of conformal geometric algebra;

[0081] Step 2: The kinematic equations of the parallel machining robot are obtained by performing the intersection operation based on the sphere geometry, and then the initial robot error model is obtained by performing the perturbation operation.

[0082] Step 3: Express the robot's passive joint errors as a function of kinematic error parameters and driving quantities. Combined with the initial robot error model, a complete robot error model including passive joint errors is obtained.

[0083] Step 4: Use a laser tracker to measure the error of the end position of the parallel machining robot to obtain the actual position of the robot and input the actual position data into the complete robot error model;

[0084] Step 5: Taking the deviation between the actual and nominal robot poses as the optimization target, the regularized Levenberg-Marquardt algorithm is used to identify the error parameters in the complete robot error model, and finally the kinematic parameters are compensated.

[0085] Example 1 (2-PRU-PUR parallel robot calibration):

[0086] The spheres in the motion space of the end points of each branch in step 1 are represented as follows:

[0087] The 2PRU-PUR (P: moving pair, R: rotating pair, U: Hooke's joint) parallel robot consists of a fixed platform, a moving platform, a PUR branch and two PRU branches, such as Figure 2 As shown. Each branch of the parallel robot is in A i Point and B i Point connected to the drive pair and the dynamic platform, C i Fixed on a fixed platform and C i A i Parallel to each other.

[0088] In order to describe the motion of the 2PRU-PUR parallel robot, a fixed coordinate system {O-xyz} is established on the fixed platform, with its origin O located at the midpoint of C1C2, the x-axis coincides with OC3, the y-axis is along the direction of OC1, and the z-axis satisfies the right-hand rule. A moving coordinate system {o-uvw} is defined on the moving platform, with o located at the midpoint of B1B2, the u-axis coincides with oB3, the v-axis is along the direction of oB1, and the w-axis satisfies the right-hand rule; C i Point coordinates are expressed as (x ci ,y ci ,z ci ) indicates that A i Point coordinates are expressed as (x ai ,yai ,z ai ) indicates. Establish a local coordinate system {O Ci}(i=1,2,3) and {O Aj}(j=1~2), such as Figure 3 shown.

[0089] Based on the above description, the nominal mechanism parameters of the 2PRU-PUR parallel robot are as follows:

[0090] x ci ,y ci ,z ci Indicates C i Point position, α ci ,β ci Indicates the deflection angle of the moving secondary vector, l i represents the length of the passive branch chain, k1, k2, k3 represent the tool structure parameters, β a1 ,γ a1 Indicates the rotational direction deviation angle of the first and second branches, lb 12 ,lb 13 ,lb 23 Indicates the hinge installation distance at the moving platform, Represents the angle between B1B2 and A3B3.

[0091] In robot analysis based on conformal geometry algebra, there are five orthogonal basis vectors in the conformal geometry space, including three Euclidean orthogonal basis vectors {e1, e2, e3} and two additional orthogonal product vectors {e + ,e -}, where the two zero vectors e0 and e ∞ Indicates the origin and the point at infinity. For this parallel robot, the connection point between branch i (i = 1 to 3) and the moving platform is defined as B i (i=1-3), point B3 is obtained by the outer product of the three spheres (see Figure 4 ). Moving platform B i The distance between (i=1~3) is defined as the geometric parameter lb ij (i=1~3,j=1~3,i≠j), with B1 as the center of the sphere, lb 13 A sphere with radius S b1b3 Expressed as:

[0092]

[0093] With B2 as the center of the ball, lb 23 A sphere with radius is represented by:

[0094]

[0095] The connection point between branch i (i=1~3) and the driving pair is defined as A i (i=1~3), sphere S with A3 as center and l3 as radius a3b3 Expressed as

[0096]

[0097] The steps for establishing the initial robot error model in step 2 are:

[0098] (1) Point B3 is at the intersection of three geometric entities (spheres). Expressed as:

[0099]

[0100] Similarly, we can get the unique solution of point B3, which is expressed as follows:

[0101]

[0102] (2) The position of the robot's end effector is obtained using the vector method. The u direction is represented by the modulus of 2B3-B1-B2, the v direction is represented by the modulus of B2-B1, w is the cross product of u and v, and k i (i=1-3) represents the movement of the end effector from point B2 along u, v and w respectively. That is, the end effector position P is expressed as:

[0103] P=B2+k1u+k2v+k3w (6)

[0104] The end position P of the mechanism can be expressed as:

[0105] f(P,r m ,θ)=0 (7)

[0106] where r m The dimensional parameters of the parallel mechanism, θ represents the passive joint angle.

[0107] 2PRU-PUR parallel robot C caused by slide installation i The point position error is expressed as δx ci ,δy ci ,δz ci The moving pair vector error caused by the slide installation is expressed as parameter δα ci ,δβ ci , the passive branch link length error is expressed as parameter δl i , the tool installation position error is expressed as parameters δk1, δk2, δk3. Due to the over-constrained structure of the mechanism, the rotational direction of the first and second branches must remain parallel, so the rotational direction vector error is expressed as δβ a1 ,δγ a1Theoretically, the hinge installation position error of the moving platform is equivalent to the fixed platform installation position error, and needs to be described using 9 parameters. However, when using a measuring instrument with the same level of accuracy as a laser tracker to calibrate the parameters of the mechanism, the U-type structural parameter error has a limited impact on the calibration results. Therefore, the hinge installation position error of the moving platform can be expressed as the distance between the three hinge center points in lb. 12 ,lb 13 ,lb 23 The driving sub-zero error can be i Point position error δz ci Therefore, no modeling is required. When designing the mechanism theoretically, it satisfies the constraint relationship A3B3·B1B2=0. Considering the influence of manufacturing error, the angular constraint error between B1B2 and A3B3 is expressed as

[0108] (3) By taking the total differential of the position equation of point B1 in the forward kinematic analysis, the mapping relationship between the position error of point B1 and the structural parameter error is obtained, which is expressed as follows:

[0109]

[0110] in They represent the differentials of the position equation of point B1 with respect to its position, structural parameters, and passive joint rotation angle; δr m Indicates the error of the structural parameters of the mechanism; δp b1 represents the position error of point B1; δθ represents the rotation angle error of the passive joint.

[0111] Formula (8) is organized into a matrix form and expressed as:

[0112]

[0113] in

[0114] Similarly, by taking the total differential of the position equations of points B1 and B2, we can obtain the error mapping relationship as follows:

[0115]

[0116] in

[0117]

[0118] (4) By differentiating the mechanism kinematic equation (6), the initial robot error model is obtained as follows:

[0119]

[0120] in Respectively represent the end position equation of the mechanism with respect to its own position and the moving platform B i Differentiation of point position and actuator structural parameters;

[0121] δp bi =[δx bi ;δy bi ;δz bi ](i=1~3) represents B i Point position error; δk i =[δk1; δk2; δk3] represents the actuator structural parameter error.

[0122] Formula B i Substitute the point position error into equation (8) and express it in matrix form as follows:

[0123]

[0124] The initial robot error model without passive joint errors can be obtained by rearranging formula (12):

[0125]

[0126] in δp is the end pose error vector, δr m is the structural parameter error vector, and δθ is the passive joint error vector.

[0127] The process of establishing the complete robot error model in step 3 is as follows:

[0128] According to the geometric constraints of the robot, the projection of the connecting rod A3B3 on the plane where the moving platform is located cannot rotate relative to B1B2, that is, the angle between B1B2 and A3B3 is a constant, set as Therefore, a single variable input-output equation about θ can be established as follows:

[0129]

[0130] Since the motion error of the passive joint is caused by the error in the dimensional parameters of the mechanism, the motion error of the passive joint can be mapped to the error in the dimensional parameters of the mechanism through the geometric relationship of the mechanism. By taking the total differential of Equation (14), the motion error of the passive joint is mapped to the error in the dimensional parameters of the mechanism, which is expressed as follows:

[0131]

[0132] in is the structural parameter error vector after mapping, is the passive joint error mapping matrix. Combined with formula (13), the mapping error model is as follows:

[0133]

[0134] where δp=[δx p ;δy p ;δz p ] represents the position error of the end effector position P; δr + Indicates the error of the structural parameters of the mechanism; is the error mapping matrix containing the passive joint error terms.

[0135]

[0136] The specific method of step 4 is:

[0137] A kinematic calibration experimental platform is established for the 2PRU-PUR parallel machining robot. Figure 5 As shown; including: 2PRU-PUR parallel robot and its control cabinet, Leica AT901-MR laser tracker measurement system, and computer control system.

[0138] The laser tracker measurement system mainly includes the host, extended base, AT Controller 900 controller, temperature and pressure sensors, which can perform real-time sensing and monitoring of the environment in which the laser tracker is located.

[0139] Accessories (all existing technologies) include RRR1.5 target ball, Leica T-Mac, and PloyWorks software.

[0140] The actual position and posture of the end effector of the parallel mechanism in the measurement coordinate system is measured using T-Mac, and the actual position and posture of the end effector in the mechanism coordinate system is obtained by combining the coordinate system transformation matrix.

[0141] The specific method of step five is:

[0142] Will and δp stacked into and δp + , and then solve the identification model Error of structural parameters δr + The optimal solution is to find the appropriate δr + Let Minimum.

[0143] Because there may be In the case of insufficient rank, the use of nonlinear least squares fitting will lead to divergence of the identification results. Regularized nonlinear least squares method can effectively prevent divergence in the identification process. Therefore, the Levenberg-Marquardt (LM) algorithm is selected to iteratively identify the structural parameter errors. The structural parameter iterative identification formula based on the LM algorithm can be expressed as:

[0144]

[0145] Where λ represents the regularization coefficient. Choosing an appropriate λ value can enhance the robustness of the identification results.

[0146] Calibration experiment considering passive joint errors:

[0147] The positioning planes are connected, the z-axis vector is obtained by fitting the data points along the slide direction of the positioning plane, and the x-axis vector is obtained by the cross product of the y-axis vector and the z-axis vector. The homogeneous transformation matrix of the measurement coordinate system relative to the fixed coordinate system is calculated.

[0148] The actual position and posture of the end effector of the parallel robot in the measurement coordinate system is measured by T-Mac, and the actual position and posture of the end effector in the robot coordinate system is obtained by combining the coordinate system transformation matrix. Figure 6 As shown by the dotted curve in , before kinematic calibration, the maximum and average values of the robot's position error are 5.3139 mm and 5.0199 mm, respectively, and the maximum and average values of the attitude error are 0.0147 rad and 0.0088 rad, respectively. Figure 6 As shown in the solid curve, after kinematic calibration, the maximum and average values of the robot's position error are reduced to 0.1749 mm and 0.1017 mm, and the maximum and average values of the attitude error are reduced to 0.0017 rad and 0.0010 rad.

[0149] The experimental results show that after calibration, the maximum value of the parallel robot's position error is reduced by 96.82% and the average value is reduced by 97.93%. The maximum value of the posture error is reduced by 87.07% and the average value is reduced by 86.36%, which verifies the effectiveness of the kinematic calibration method proposed in this invention.

Claims

1. A kinematic calibration method for a parallel machining robot considering passive joint errors is performed as follows: Step 1: Based on the geometric structure of the parallel machining robot, a sphere representing the motion space of each branch end point is constructed within the framework of conformal geometric algebra; Step 2: The kinematic equations of the parallel machining robot are obtained by performing the intersection operation based on the sphere geometry, and then the initial robot error model is obtained by performing the perturbation operation. Step 3: Express the robot's passive joint errors as a function of kinematic error parameters and driving quantities. Combined with the initial robot error model, a complete robot error model including passive joint errors is obtained. Step 4: Use a laser tracker to measure the error of the end position of the parallel machining robot to obtain the actual position of the robot and input the actual position data into the complete robot error model; Step 5: Taking the deviation between the actual and nominal robot poses as the optimization target, the regularized Levenberg-Marquardt algorithm is used to identify the error parameters in the complete robot error model, and finally the kinematic parameters are compensated.

2. The kinematic calibration method of a parallel robot considering passive joint errors according to claim 1, characterized in that: The spheres in the motion space of the end points of each branch in step (1) are as follows: In robot analysis based on conformal geometry algebra, there are five orthogonal basis vectors in the conformal geometry space, including three Euclidean orthogonal basis vectors {e1, e2, e3} and two additional orthogonal product vectors {e + ,e-}, where the two zero vectors e0 and e ∞ Indicates the origin and the point at infinity; for the parallel processing robot, the connection point of branch i (i = 1 to 3) and the moving platform is defined as Bi (i = 1 to 3), and point B3 is obtained by the outer product of the three spheres; the distance between the moving platforms Bi (i = 1 to 3) is defined as the geometric parameter lbij (i = 1 to 3, j = 13, i ≠ j), with B1 as the center of the sphere, lb 13 A sphere with radius S b1b3 Expressed as: With B2 as the center of the ball, lb 23 A sphere with radius is represented by: The connection point between branch i (i=1-3) and the driving pair is defined as Ai (i=1-3). The sphere with A3 as the center and l3 as the radius is expressed as: 。 3. The kinematic calibration method of a parallel robot considering passive joint errors according to claim 2, characterized in that: The initial robot error model in step (2) is: Where: δp is the end effector position error, represents the partial derivative of the position equation F with respect to the end effector position p, Represents the position equation F versus the structural parameter r m The partial derivative of δr m is the structural parameter error, represents the partial derivative of the position equation F with respect to the passive joint angle θ, and δθ is the passive joint angle error.

4. The kinematic calibration method of a parallel robot considering passive joint errors according to claim 3, characterized in that: The steps for establishing the initial robot error model in step (2) are: (1) Point B3 is at the intersection of three spheres. Expressed as: The only solution for point B3 is: (2) The position of the robot's end effector is obtained using the vector method. The u-axis direction of the moving coordinate system is represented by the modulus of 2B3-B1-B2, the v-axis direction is represented by the modulus of B2-B1, and the w-axis is the cross product of u and v. ki (i = 1 to 3) represents the movement of the end effector from point B2 along u, v, and w, respectively. That is, the end effector position p is expressed as: p=B2+k1u+k2v+k3w (7) The end effector position p of the mechanism is expressed as a function of the position equation F: F(p,r m ,θ)=0 (8) where r m Structural parameters of the parallel mechanism, θ represents the passive joint angle; (3) By taking the total differential of the position equation of point B1 in the forward kinematic analysis, the mapping relationship between the position error of point B1 and the structural parameter error is obtained, which is expressed as: in They represent the differentials of the position equation of point B1 with respect to its position, structural parameters, and passive joint rotation angle; δp b1 Indicates the position error of point B1; δr m represents the structural parameter error; δθ represents the passive joint angle error; Formula (8) is organized into a matrix form and expressed as: in Similarly, by taking the total differential of the position equations of points B1 and B2, we can obtain the error mapping relationship as follows: in δp b2 Indicates the position error of point B2, They represent the differentials of the position equation of point B2 with respect to its position, structural parameters, and passive joint rotation angle respectively; δp b3 Indicates the position error of point B3, They represent the differentials of the position equation of point B3 with respect to its position, structural parameters, and passive joint rotation angle respectively; (4) By differentiating the mechanism kinematic equation (6), the initial robot error model is obtained as follows: in Respectively represent the position equation F about its own position and the moving platform B i Differentiation of point position and actuator structural parameters; δp bi =[δx bi ;δy bi ;δz bi ](i=1~3) represents B i Point position error; δk i =[δk1; δk2; δk3] represents the actuator structural parameter error; Formula B i Substituting the point position error into equation (8), it is written in the following matrix form: in Express the position equation F with respect to the actuator structural parameter k i The differential of Represents the structural parameter r k The actuator structural parameter error k i composition, The initial robot error model without passive joint errors can be obtained by rearranging formula (12): in: They represent the position equation F with respect to the structural parameter r km , the passive joint angle θ, and the differential of the end effector position p.

5. The kinematic calibration method of a parallel robot considering passive joint errors according to claim 4, characterized in that: The process of establishing the complete robot error model in step (3) is as follows: According to the geometric constraints of the robot, define the geometric angle between the link B1B2 and the link A3B3 is an invariant; therefore, a single variable input-output equation about the passive joint angle θ is established, which is expressed as: Where l3 is the length of the connecting rod A3B3 of the mechanism. Since the motion error of the passive joint is caused by the error of the dimensional parameters of the mechanism, the motion error of the passive joint is mapped to the error of the dimensional parameters of the mechanism through the geometric relationship of the mechanism. By taking the total differential of Equation (14), the passive joint error is mapped to the function form of the kinematic error parameter and the driving amount, which is expressed as: in is the structural parameter error after mapping, is the error mapping matrix including the passive joint error; combined with formula (13), the complete robot error model including the passive joint error is obtained: where δp=[δx p ;δy p ;δz p ] represents the position error of the end effector; δr + Represents the structural parameter error after mapping; is the error mapping matrix containing the passive joint errors.

6. The kinematic calibration method of a parallel robot considering passive joint errors according to claim 5, characterized in that: The specific method of step (4) is: The actual position and orientation of the end effector of the parallel mechanism in the measurement coordinate system is measured using T-Mac. The actual position and orientation of the end effector in the mechanism coordinate system is obtained by combining the coordinate system transformation matrix. The measured actual position and orientation is used as the input of the error model. Considering the current structural parameters In the case of the mechanism forward kinematics solution, the theoretical position p of the end effector is obtained (k) , the end pose p measured by the laser tracker + ; Combined with the error model, the error identification matrix under the current structural parameters is obtained 7. The kinematic calibration method of a parallel robot considering passive joint errors according to claim 6, characterized in that: The derivation process of the structural parameter iterative identification formula used for identification in step (5) is: (1) and δp stacked into and δp + , and then solve the identification model Error of structural parameters δr + The optimal solution, that is, to find the appropriate δr + Let smallest; (2) Taking the deviation between the actual position and the nominal position of the robot as the optimization target, the regularized Levenberg-Marquardt algorithm is selected to iteratively identify the error parameters; the formula for iterative identification of structural parameters based on the regularized Levenberg-Marquardt algorithm is expressed as: Where λ represents the regularization coefficient.

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