Kinematics calibration and measurement pose optimization method for motion redundancy parallel mechanism

Abstract

The invention relates to the technical field of robot control, in particular to a kinematics calibration and measurement pose optimization method for a motion redundancy parallel mechanism, which comprises the following steps of: performing error modeling including redundant error parameters on the basis of inverse kinematics of a motion redundancy parallel mechanism, and then designing a parameter identification algorithm to obtain an optimal measurement pose. According to the method, the divergence problem in the identification process is effectively prevented, and the influence of the measurement error on the identification result is reduced as much as possible.

Description

Technical Field

[0001] The present invention relates to the technical field of robot control, and specifically to a kinematic calibration and measurement pose optimization method for a motion-redundant parallel mechanism. Background Technique

[0002] In parallel mechanisms, according to the specific type of redundancy, they can be divided into three types: redundant drive, motion redundancy, and hybrid redundancy. In a motion-redundant parallel mechanism, the number of drives of the mechanism is equal to the degrees of freedom of the mechanism, but the degrees of freedom of the moving platform are less than those of the entire mechanism, which results in the mechanism having local degrees of freedom, that is, the so-called redundant degrees of freedom. This redundant design enables the mechanism to have unique advantages in aspects such as performance optimization, singularity avoidance, and fault tolerance. By adding motion-redundant degrees of freedom to the branch chains, it helps to avoid the singular positions of the mechanism, thereby expanding the pose workspace. These advantages such as good flexibility, large reachable workspace, and avoiding singular configurations make it have great market application prospects in aspects such as remote surgery, material sorting and handling, micro-operation and assembly, and human-robot collaborative operation. However, the redundant drives also bring complex kinematic problems and increase the difficulty of model control and precision measurement.

[0003] Kinematic calibration is a key step to improve the positioning accuracy of robots. Through accurate kinematic parameter identification, manufacturing and assembly errors can be compensated, and the overall performance of the mechanism can be improved. For motion-redundant parallel mechanisms, the redundant drives introduce more joints and linkages, making the kinematic model more complex and the number of parameters significantly increase. Therefore, studying an efficient kinematic calibration method applicable to such mechanisms is crucial for ensuring their high-precision operation.

[0004] For traditional non-redundant parallel mechanisms, since each drive in their kinematic model has a unique solution, only the geometric parameters of the mechanism, such as link lengths, joint positions, etc., need to be accurately measured and adjusted to ensure that the theoretical model is consistent with the actual behavior. Their calibration is relatively simple and mainly focuses on improving mechanical accuracy and reducing errors. However, the calibration of motion-redundant mechanisms is more complex and also needs to comprehensively consider redundant degrees of freedom to achieve the best dynamic performance and operation efficiency. Summary of the Invention

[0005] Aiming at the deficiencies of the existing technology, the purpose of the present invention is to provide a kinematic calibration and measurement pose optimization method for a motion-redundant parallel mechanism.

[0006] To achieve the above purpose, the present invention provides the following technical solution: A kinematic calibration and measurement pose optimization method for a motion-redundant parallel mechanism, and its steps are as follows:

[0007] (1) Solve the inverse kinematic expression of the motion-redundant parallel mechanism through the closed-loop vector method;

[0008] (2) On the basis of step (1), analyze the error source parameters and establish a kinematic error model for the motion-redundant parallel mechanism;

[0009] (3) Design a parameter identification algorithm that combines the characteristics of the gradient descent method and the Gauss-Newton method. Solve the forward kinematics iteratively by the numerical method, and complete the identification process by optimizing the residual between the actual pose and the theoretical pose;

[0010] (4) Design an optimal measurement pose algorithm to optimize the measurement pose and reduce the interference of errors on the identification result;

[0011] (5) After completing the above steps, perform parameter identification on the optimized measurement pose, and then substitute the identified error parameters into the control model for correction to achieve compensation for the end pose.

[0012] In some embodiments, the motion-redundant parallel mechanism is composed of three branches connecting the moving platform and the fixed platform. Two of the branches are motion-redundant PUPR branches, which are symmetrically arranged in structure and are connected and fixed to the fixed platform through guide rails. The last branch is a motion-redundant PRPU branch, which is fixed to the fixed platform through a guide rail. Each branch uses two P pairs as driving units to adjust the pose of the end of the moving platform. Among them, the P pair on the guide rail slides with a slider, and the other P pair expands and contracts through an electric cylinder.

[0013] In some embodiments, according to step (1), the specific method is as follows:

[0014] Solve the inverse kinematics by the closed-loop vector method, where the closed-loop vector equation of the mechanism is:

[0015] q i =P m +b i -a i i=(1,2,3) (1)

[0016] The inverse kinematics solution of the parallel robot obtained by the closed-loop vector method is expressed as:

[0017]

[0018] In some embodiments, according to step (2), the specific method is as follows: Perform kinematic calibration on the mechanism and establish an error model.

[0019] First, represent the rotation matrix of the mechanism as:

[0020]

[0021] Include all the structural parameters corresponding to the above error parameters into the inverse solution expression to obtain:

[0022]

[0023] Wherein:

[0024] L = (zc1 - zp + xla1sβ - zla1cαcβ + yla1cβsα),

[0025] M = (xla1cβ - xc1 + zptβ + zla1cαsβ - yla1sαsβ),

[0026] N = (q4 + yc1 - yla1cα - zla1sα),

[0027] O = (zp - zc2 - xla2sβ + zla2cαcβ + yla2cβsα),

[0028] P = (xla2cβ - xc2 + zptβ + zla2cαsβ + yla2sαsβ), Q = (q5 + yc2 - yla2cα + zla2sα),

[0029] R = (zp - sβ(xc3 + xla3) + cαcβ(zc3 + zla3) + cβsα(yc3 + yla3)),

[0030] S = (cβ(xc3 + xla3) - q6 + zptβ + cαsβ(zc3 + zla3) + sαsβ(yc3 + yla3)),

[0031] T = (cα(yc3 + yla3) - sα(zc3 + zla3)).

[0032] Further arranging the above formula gives:

[0033]

[0034] Performing total differentiation on both sides of Equation (5) gives:

[0035]

[0036] Wherein:

[0037]

[0038] Solving the system of linear equations in three variables for Equation (6), where Δε1 0 = (Δl ai Δq4Δc i Δq1) ú is a parameter of the equation, and δα, δβ, δz p are unknowns. Writing the equation in matrix form gives:

[0039]

[0040] In the formula, let

[0041]

[0042] The mapping relationship between the end pose error and the kinematic error parameters is obtained as follows:

[0043]

[0044] Usually, a tool is installed at the end of the moving platform of a parallel mechanism. Let the length of the tool at the end of the moving platform of the mechanism be H. Then, the coordinates of the end point of the tool are expressed in the moving coordinate system o-uvw as: H o =(0 0 H) T , According to the coordinate transformation, its expression in the fixed coordinate system O-xyz is obtained as:

[0045]

[0046] where, P m is obtained from the self-constraint conditions of the mechanism:

[0047] P m =(z p tanβ0 z p ) T (10)

[0048] Performing a first-order perturbation on both sides of Equation (10) gives:

[0049] Δr=(δz p tanβ+z p δαsec 2 β0δz p ) T (11)

[0050] Converting Equation (11) into matrix form gives:

[0051]

[0052] From the mapping relationship between the skew-symmetric matrix of angular velocity and the rotation matrix the angular velocity of the moving platform is obtained:

[0053] ΔΩ=(cosβδα δβ - sinβδα) T (13)

[0054] The relationship between the position of the end point of the tool and the pose of the moving platform is:

[0055] p = r + R p l c (14)

[0056] Perform a first-order perturbation on both sides of Equation (14) to obtain:

[0057] Δp = Δr + ΔΩ × (R p l c ) + R p Δl c (15)

[0058] where Δp = (Δx p Δy p Δz p ) T represents the position deviation of the end effector in the fixed coordinate system, and Δl c = (δd1 δd2 δd3) T refers to the tool error of the end effector in the moving coordinate system;

[0059] Substitute Equation (15) into Equation (12) to obtain:

[0060]

[0061] Further arrangement gives:

[0062] Δp i = J i Δε i (i = 1, 2, 3) (17)

[0063] In the formula: J d The value of is:

[0064]

[0065] Thus, the establishment of the error model of this kinematically redundant parallel mechanism is completed.

[0066] In some of these embodiments, according to step (3), the parameter identification steps are as follows:

[0067] (1) Before starting iterative identification, based on the theoretical structural parameters r k Select measured pose data in the workspace of the 2PUPR - PRPU kinematically redundant parallel mechanism for error parameter identification. Substitute the measured pose into the inverse solution expression to solve for the corresponding drive values. Considering the current structural parameters , use the forward kinematics of the mechanism to solve for the theoretical pose T (k) of the end effector, and combine with the error model to obtain the error identification matrix under the current structural parameters

[0068] (2) Let the error parameter of the mechanism be δr k , and the actual parameter of the mechanism is obtained as r k+δr k , use the true solution to obtain the pose T of the mechanism end with structural parameter errors k , and obtain the actual measured pose T by adding measurement noise m ;

[0069] (3) Subtract T m -T (k) to obtain the pose error δT (k) , and then use the regularized non-linear least squares method to identify the normalized identification matrix to obtain the error parameters of the mechanism

[0070] (4) Then judge whether δT (k) -δT (k-1) is less than the identification accuracy index. If the index is not satisfied, update the structural parameters to Repeat the iterative process until the pose error converges, and finally output the identified structural parameters and structural parameter errors.

[0071] In some of these embodiments, the parameter identification iteration formula is expressed as:

[0072] δr k (k + 1) = δr k (k) + ((J k ) T J k + λI) -1 (J k ) T δT k (19)

[0073] where λ represents the regularization coefficient.

[0074] In some of these embodiments, according to step (4), the specific method is as follows:

[0075] Before optimizing the measurement points, first pre-select the measurement points. The measurement points should cover the working space as much as possible and include the measurement points on the boundary of the working space.

[0076] After pre-selecting the measurement poses, use the measurement pose optimization algorithm to optimize the pre-selected measurement points.

[0077] By introducing the observability index to evaluate the quality of the measurement poses, the observability index is used as the index for pose optimization and is associated with the identification matrix. There are mainly the following types:

[0078]

[0079]

[0080] O3 = σm (22)

[0081]

[0082]

[0083]

[0084] In the formula, n represents the number of measurement points, and σ1≥σ2≥…≥σ m represents the non-zero singular values of the identification matrix composed of n measurement poses;

[0085] By comparing the differences of each observability index and its influence on the uncertainty of the end pose, O3 is determined, that is, the minimum non-zero singular value of the identification matrix is the optimal observability index. Taking O3 as the optimization goal, the pose data set is optimized through an optimization algorithm.

[0086] In some of these embodiments, a locust optimization algorithm is set, and its mathematical formula is:

[0087] X i =S i +G i +A i (26)

[0088] where X i is the position of the locust, S i is the mutual influence force between individuals, G i is the gravity of the locust, and A i is the wind force received by the locust;

[0089] The social role of the locust has the greatest influence on the locust's position, and the expression is as follows:

[0090]

[0091] where d ij =|x i -x j | is the distance between the i-th locust and the j-th locust; d ij =(x j -x i ) / d ij is the unit vector; s is the calculation method of the social force as follows:

[0092]

[0093] In the formula, f is the attraction intensity, l is the attraction intensity scale, and f = 0.5, l = 1.5 are taken.

[0094] Ignoring the influence of gravity and wind force, X i can be expressed as:

[0095]

[0096] In some of these embodiments, the locust optimization algorithm is used as the preferred pose measurement algorithm, and its steps for preferentially measuring the pose are as follows:

[0097] (1) Initialize parameters, including initial parameters such as the locust population size and the number of algorithm iterations;

[0098] (2) Each locust represents a pose subset. Calculate the fitness of each locust and the minimum singular value of the identification matrix;

[0099] (3) Calculate the current best fitness and maximize the minimum singular value;

[0100] (4) Update the position of the locusts, calculate the fitness after the update, compare it with that before the update, and determine whether an update is needed;

[0101] (5) Determine whether the number of iterations has been reached. If so, the steps end, and the position of the current global optimal solution is the optimal solution.

[0102] In some of these embodiments, according to step (5), the specific method is as follows:

[0103] Calculate the drive values corresponding to each drive joint by taking the inverse solution of the nominal end pose, and then substitute the identified structural error parameter r k (k) and the identified Jacobian matrix at the current pose into the solution to obtain the pose error T to be corrected k (k) Among them, the kinematic parameters after error compensation are expressed as:

[0104]

[0105] Substitute the compensated kinematics into the inverse solution expression for solution to obtain the corrected drive values of the parallel mechanism, and input the above corrected drive values into the control system to complete the compensation of the end pose.

[0106] Compared with the prior art, the beneficial effects of the present invention are as follows: By establishing an accurate kinematic error model for a motion-redundant parallel mechanism, the accuracy and reliability of the motion-redundant parallel mechanism in actual operation can be improved after kinematic calibration. This model includes the error parameters of the redundant branches, which, due to its structural design, can perform motions in degrees of freedom that are inaccessible in traditional non-redundant parallel mechanisms. During the entire kinematic calibration process, the influence of the redundant branch parameters is comprehensively considered, and through a measurement pose optimization algorithm, the influence of measurement errors is reduced, providing a better data set for parameter identification, improving its identification effect, providing theoretical guidance for the kinematic calibration of the motion-redundant parallel mechanism, and ultimately improving the control accuracy and positioning accuracy of the motion-redundant parallel mechanism through error compensation.

[0107] Details of one or more embodiments of the present application are set forth in the following drawings and description to make other features, objects, and advantages of the present application more concise and understandable. The present application will be described in detail and understood through the embodiments of the present application. BRIEF DESCRIPTION OF THE DRAWINGS

[0108] Figure 1 is a structural schematic diagram of a 2PUPR-PRPU motion-redundant parallel mechanism;

[0109] Figure 2 is a closed-loop vector diagram of the structure of a 2PUPR-PRPU motion-redundant parallel mechanism;

[0110] Figure 3 is a flow chart of parameter identification simulation;

[0111] Figure 4 is a flow chart of numerical calculation of forward kinematics;

[0112] Figure 5 is a flow chart of the measurement pose optimization algorithm;

[0113] Figure 6 is an input compensation block diagram for error compensation. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0114] The structure of traditional non-redundant mechanisms is relatively simple, and the number of control drives is also relatively small. However, for motion-redundant parallel mechanisms, due to the large number of drives, the control difficulty increases significantly. In terms of kinematic calibration, the research on kinematic calibration of traditional non-redundant mechanisms has tended to saturation, and various modeling methods and algorithms have been relatively perfect. For motion-redundant parallel mechanisms, there is less research on kinematic calibration. Although the methods of non-redundant parallel mechanisms can be used, due to the characteristics of motion-redundant parallel mechanisms, it is determined that they will be more complex in error modeling, error measurement, parameter identification, and error compensation. Therefore, how to apply the motion-redundant characteristics to kinematic calibration is the key to kinematic calibration of motion-redundant parallel mechanisms.

[0115] This application aims to address the deficiencies in the kinematic calibration research of motion-redundant parallel mechanisms in the existing technology. Currently, for traditional non-redundant parallel mechanisms, the kinematic calibration research is relatively complete, but there are relatively few literatures and studies on the calibration methods for motion-redundant parallel mechanisms. This application specifically focuses on the characteristics of motion-redundant parallel mechanisms and conducts kinematic calibration research to fill the gaps in the existing research. To this end, this application proposes a kinematic calibration method for motion-redundant parallel mechanisms. First, based on the inverse kinematics of the motion-redundant parallel mechanism, an error model including redundant error parameters is established. Then, the parameter identification algorithm is designed. The LM algorithm is selected to iteratively identify the structural parameter errors, which can effectively prevent divergence problems during the identification process. In addition, a measurement pose optimization algorithm is designed to minimize the influence of measurement errors on the identification results as much as possible.

[0116] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0117] As Figure 1 , the present invention provides a technical solution: a kinematic calibration and measurement pose optimization method for a motion-redundant parallel mechanism, using a 2PUPR-PRPU motion-redundant parallel mechanism as the case implementation mechanism of the present invention. The implementation steps are as follows:

[0118] 1. Solve the inverse kinematics expression of the motion-redundant parallel mechanism by the closed-loop vector method;

[0119] 2. On the basis of step 1, analyze the error source parameters and establish a kinematic error model for the motion-redundant parallel mechanism. This model can accurately represent the mapping relationship between the end pose error and the kinematic error parameters;

[0120] 3. Apply a parameter identification algorithm that can correlate the error parameters of the motion-redundant branches. This algorithm combines the characteristics of the gradient descent method and the Gauss-Newton method, and numerically iteratively solves the forward kinematics. The forward kinematics solution is a pose composed of multiple groups of data. Finally, the identification process is completed by optimizing the residual between the actual pose and the theoretical pose;

[0121] 4. The influence of measurement errors on the identification results varies at different poses. Therefore, design a measurement pose optimization algorithm to optimize the measurement poses to minimize the interference of these errors on the identification results;

[0122] 5. After completing the above steps, identify the parameters of the preferred measurement pose, and then substitute the identified error parameters into the control model for correction to achieve compensation for the end pose.

[0123] In the above solution, the controlled 2PUPR-PRPU motion-redundant parallel structure is composed of three branches connecting the moving platform and the fixed platform. Among them, Branch 1 and Branch 2 belong to the motion-redundant PUPR branches, while Branch 3 is a motion-redundant PRPU branch. Branch 1 and Branch 2 are symmetrically arranged in structure and are connected and fixed to the fixed platform through guide rails. Branch 3 is also fixed to the fixed platform through a guide rail. Each branch uses two P pairs as driving units to adjust the pose of the end of the moving platform. Among them, the P pair on the guide rail slides with a slider, and the other P pair expands and contracts through an electric cylinder.

[0124] As Figure 2 shown, the inverse kinematics is solved by the closed-loop vector method, and the closed-loop vector equation of the mechanism is:

[0125] q i = P m + b i - a i i = (1, 2, 3) (1)

[0126] Specifically, in this implementation case, the inverse kinematics of the 2PUPR-PRPU parallel robot obtained by the closed-loop vector method is expressed as:

[0127]

[0128] As can be seen from Equation (2), when the pose of the moving platform α, β, z p is determined, each drive q i (i = 1, 2,..., 6) of the 2PUPR-PRPU motion-redundant parallel mechanism is not a unique value, which is also a common feature of the motion-redundant parallel mechanism, that is, there are multiple solutions for the drive. Only when the input variables of the mechanism α, β, z p , q4, q5, q6 are all determined, can q i (i = 1, 2, 3) be obtained as the unique solution.

[0129] To perform kinematic calibration on this mechanism, an error model must be established first. Through error source analysis, the following error parameters are determined: δxl ai , δyl ai , δzl ai (rod length errors of the moving platform in the x, y, and z directions), δq 4,5,6 (zero position errors of the three drives of the fixed platform), δxc i , δyc i , δzc i(Position error of the axes of the two rotating shafts of the universal hinge in the x, y, and z directions), δd1, δd2, δd3 (tool errors in the x, y, and z directions).

[0130] First, the rotation matrix of the mechanism is expressed as:

[0131]

[0132] All the structural parameters corresponding to the above error parameters are included in the inverse solution expression to obtain:

[0133]

[0134] Where:

[0135] L = (zc1 - zp + xla1sβ - zla1cαcβ + yla1cβsα), M = (xla1cβ - xc1 + zptβ + zla1cαsβ - yla1sαsβ),

[0136] N = (q4 + yc1 - yla1cα - zla1sα), O = (zp - zc2 - xla2sβ + zla2cαcβ + yla2cβsα),

[0137] P = (xla2cβ - xc2 + zptβ + zla2cαsβ + yla2sαsβ), Q = (q5 + yc2 - yla2cα + zla2sα),

[0138] R = (zp - sβ(xc3 + xla3) + cαcβ(zc3 + zla3) + cβsα(yc3 + yla3)),

[0139] S = (cβ(xc3 + xla3) - q6 + zptβ + cαsβ(zc3 + zla3) + sαsβ(yc3 + yla3)),

[0140] T = (cα(yc3 + yla3) - sα(zc3 + zla3)).

[0141] Further arranging Equation (4) gives:

[0142]

[0143] Taking the total differential of both sides of the above equation gives:

[0144]

[0145] Where:

[0146]

[0147] Solve the system of linear equations of three variables for Equation (6), where are the equation parameters, δα, δβ, δz p are the unknowns. Writing the equation in matrix form gives:

[0148]

[0149] In the formula, let

[0150]

[0151] Thus, the mapping relationship between the end - pose error and the kinematic error parameters can be obtained as:

[0152]

[0153] Usually, a tool is installed at the end of the moving platform of a parallel mechanism. Let the length of the tool at the end of the moving platform of the mechanism be H. Then the coordinates of the end - point of the tool can be expressed in the moving coordinate system o - uvw as: H o =(0 0 H) T . According to coordinate transformation, its expression in the fixed coordinate system O - xyz can be obtained as:

[0154]

[0155] where, P m is obtained from the self - constraint conditions of the mechanism:

[0156] P m =(z p tanβ0 z p ) T (10)

[0157] Performing a first - order perturbation on both sides of Equation (10) gives:

[0158] Δr=(δz p tanβ + z p δαsec 2 β0δz p ) T (11)

[0159] Converting Equation (11) into matrix form gives:

[0160]

[0161] From the mapping relationship between the skew - symmetric matrix of angular velocity and the rotation matrix the angular velocity of the moving platform can be obtained as:

[0162] ΔΩ=(cosβδα δβ - sinβδα) T (13)

[0163] The relational expression between the position of the tool tip and the pose of the moving platform is:

[0164] p = r + R p l c (14)

[0165] Performing a first-order perturbation on both sides of Equation (14), we obtain:

[0166] Δp = Δr + ΔΩ × (R p l c ) + R p Δl c (15)

[0167] where Δp = (Δx p Δy p Δz p ) T represents the position deviation of the end tool in the fixed coordinate system, and Δl c = (δd1 δd2 δd3) T refers to the tool error of the end tool in the moving coordinate system.

[0168] Substituting Equation (15) into Equation (12), we can obtain:

[0169]

[0170] After further arrangement, we can obtain:

[0171] Δp i = J i Δε i (i = 1, 2, 3) (17)

[0172] In the formula: J d The value of is:

[0173]

[0174] Thus, the establishment of the error model of this kinematically redundant parallel mechanism is completed.

[0175] After completing the kinematic error modeling, it is necessary to design a parameter identification algorithm. The LM algorithm is a parameter identification algorithm that combines the characteristics of the gradient descent method and the Gauss-Newton method. As Figure 3 is the flowchart of parameter identification, and the specific steps are as follows:

[0176] 1. Before starting the iterative identification, based on the theoretical structural parameters r kMeasure the pose data in the workspace of the 2PUPR-PRPU motion redundant parallel mechanism to identify the error parameters. Substitute the measured pose into the inverse solution expression to solve for the corresponding drive values. Considering the current structural parameters in this case, use the forward kinematics of the mechanism to solve for the theoretical pose T (k) of the end effector, and combine with the error model to obtain the error identification matrix under the current structural parameters

[0177] 2. Let the error parameter of the mechanism be δr k , and the actual parameter of the mechanism can be obtained as r k +δr k . Use the forward solution to find the pose of the end of the mechanism with structural parameter errors T k , and obtain the actual measured pose T m by adding measurement noise.

[0178] 3. Subtract T m from T (k) to obtain the pose error δT (k) . Then use the regularized nonlinear least squares method to identify the normalized identification matrix to obtain the error parameters of the mechanism Next, judge whether δT (k) minus δT (k-1) is less than the identification accuracy index. If this index is not satisfied, update the structural parameters to Repeat the iteration process until the pose error converges, and finally output the identified structural parameters and structural parameter errors.

[0179] Among them, the parameter identification iteration formula based on the LM algorithm is expressed as:

[0180] δr k (k + 1) = δr k (k) + ((J k ) T J k + λI) -1 (J k ) T δT k (19)

[0181] In the formula, λ represents the regularization coefficient. Selecting an appropriate regularization coefficient can increase the accuracy and robustness of the identification results.

[0182] As Figure 4 shown is the numerical calculation flowchart of the forward kinematics solution. It should be noted that since each pose of the 2PUPR-PRPU motion redundant parallel mechanism corresponds to q i(i = 4, 5, 6) are all variable. Therefore, when solving the forward kinematics, the pose iteration solution also needs to be carried out according to each group of determined q i (i = 1, 2, …, 6).

[0183] Parameter identification is an identification model constructed based on the measured end - pose error. Therefore, measurement of calibration points is required. Due to the influence of the environment and equipment, there are measurement errors in the measured data, and these measurement errors will have a certain impact on the parameter identification results. Therefore, it is necessary to optimize the measurement points to reduce the influence of measurement errors on the parameter identification results. Before optimizing the measurement points, the measurement points need to be pre - selected first. The measurement points should cover the workspace as much as possible. Especially some measurement points at the boundary of the workspace are often suitable choices for measuring poses.

[0184] After pre - selecting the measurement poses, it is necessary to use the measurement pose optimization algorithm to optimize the poses of the pre - selected measurement points. For this purpose, several observability indices are introduced to evaluate the quality of the measurement poses. The observability index, as an index for pose optimization, is associated with the identification matrix and mainly includes the following several types:

[0185]

[0186]

[0187] O3 = σ m (22)

[0188]

[0189]

[0190]

[0191] In the formula, n represents the number of measurement points, and σ1≥σ2≥...≥σ m represents the non - zero singular values of the identification matrix composed of n measurement poses.

[0192] By comparing the differences of each observability index and its influence on the uncertainty of the end - pose, O3 is determined, that is, the minimum non - zero singular value of the identification matrix is the best observability index. Therefore, in order to obtain the optimal measurement pose set, O3 is used as the optimization goal, and the pose data set is optimized through the optimization algorithm.

[0193] The Grasshopper Swarm Optimization (GSO) algorithm is a swarm intelligence optimization algorithm that simulates the group behavior of grasshoppers. It shows strong search ability in solving global optimization problems. The population migration and foraging behaviors of grasshoppers in nature divide the search into exploration and exploitation. The mathematical formula of this algorithm is as follows:

[0194] X i = S i + G i + A i (26)

[0195] Among them, X i is the position of the locust, S i is the mutual influence between individuals, G i is the gravity of the locust 2, A i is the wind force acting on the locust.

[0196] The social role of the locust has the greatest impact on the locust position, and the expression is as follows:

[0197]

[0198] Among them, d ij = |x i - x j | is the distance between the i-th locust and the j-th locust; d ij = (x j - x i ) / d ij is the unit vector; s is the calculation method of social force as follows:

[0199]

[0200] In the formula, f is the attraction intensity, l is the attraction intensity scale, and take f = 0.5, l = 1.5.

[0201] Ignoring the influence of gravity and wind force, X i can be expressed as:

[0202]

[0203] Taking the GSO algorithm as the optimal pose measurement algorithm, as Figure 5 shown in the flowchart of the optimal pose, its steps are:

[0204] 1. Initialize parameters, including initial parameters such as the locust population size and the number of algorithm iterations;

[0205] 2. Each locust represents a pose subset, and calculate the fitness of each locust (the minimum singular value of the identification matrix);

[0206] 3. Calculate the current best fitness (maximize the minimum singular value);

[0207] 4. Update the position of the locust, calculate the fitness after update, compare it with the previous one, and judge whether to update;

[0208] 5. Determine whether the iteration count is reached. If so, the program ends, and the position of the current globally optimal solution is the optimal solution.

[0209] With the optimized measured pose, compared with the initial pose, the minimum singular value of its identification matrix is the largest globally. The larger the minimum singular value, it means that the identification matrix has better "response" or "sensitivity" in all directions, which helps to estimate the system parameters more reliably.

[0210] The last step in the kinematic calibration experiment is error compensation. Error compensation is the process of writing the identified error parameters into the control model, such as Figure 6 Input the compensation block diagram for error compensation. Calculate the drive values corresponding to each driving joint by inverse solution of the nominal pose of the end effector, and then substitute the identified structural error parameters and the identified Jacobian matrix at the current pose into the solution to obtain the pose error to be corrected In this process, the kinematic parameters after error compensation are expressed as:

[0211]

[0212] Substitute the compensated kinematics into the inverse solution expression for solution to obtain the corrected drive values of the parallel mechanism. Input these corrected drive values into the control system to complete the compensation of the end effector pose. For the 2PUPR-PRPU kinematically redundant parallel mechanism, when solving the corrected drive values, due to the existence of redundant branches, the drive values q i (i = 1, 2, 3) have multiple solutions. Therefore, give priority to compensating the zero position error identified for the drives q i (i = 4, 5, 6). Calculate the drive values of each pose drive q i (i = 4, 5, 6) after compensation, and then solve for the drive values of the drives q i (i = 1, 2, 3). Finally, input q i (i = 1, 2,..., 6) into the control system to complete the compensation of the end effector pose.

[0213] Through the proposed technical solution, by performing kinematic calibration on the motion-redundant parallel mechanism, the control accuracy of the motion-redundant parallel mechanism and the positioning accuracy of the moving platform are significantly improved. The error model containing redundant parameters can fully reflect the characteristics of motion redundancy, thereby enabling more accurate system analysis and performance optimization, and improving the dynamic response of the system. In parameter identification, the regularized nonlinear least squares method is used to identify the normalized identification matrix, which not only improves the performance of the model in the data, increases the stability and accuracy of the entire identification process, but also enhances the robustness and reliability of parameter identification, and optimizes the generalization ability of the model. By optimizing the pre-selected measurement poses, the influence of many measurement errors on parameter identification is reduced, which helps to improve the identification effect. Finally, through error compensation, the identified parameters are input into the controller, greatly improving the control accuracy and positioning accuracy of the motion-redundant parallel mechanism.

[0214] The above embodiments only represent several implementation manners of the present application. The description is relatively specific and detailed, but it should not be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application shall be subject to the appended claims.

[0215] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A kinematic calibration method for a motion-redundant parallel mechanism and an optimal selection method for measured poses, characterized in that: The steps are as follows: (1) Solve the inverse kinematic expression of the motion-redundant parallel mechanism by the closed-loop vector method; (2) On the basis of step (1), analyze the error source parameters and conduct kinematic error modeling of the motion-redundant parallel mechanism; (3) Design a parameter identification algorithm, which combines the characteristics of the gradient descent method and the Gauss-Newton method, solves the forward kinematics by numerical iteration, and completes the identification process by optimizing the residual between the actual pose and the theoretical pose; (4) Design an optimal measurement pose algorithm to optimize the measurement pose and reduce the interference of errors on the identification result; (5) After completing the above steps, perform parameter identification on the optimized measurement pose, and then substitute the identified error parameters into the control model for correction to achieve compensation for the end pose.

2. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 1, characterized in that: The described motion-redundant parallel mechanism consists of three branches connecting the moving platform and the fixed platform. Two of the branches are motion-redundant PUPR branches, which are symmetrically arranged in structure and are connected and fixed to the fixed platform through guide rails. The last branch is a motion-redundant PRPU branch, which is fixed to the fixed platform through a guide rail. Each branch uses two P pairs as drive units to adjust the pose of the end of the moving platform. Among them, the P pair on the guide rail slides with a slider, and the other P pair expands and contracts through an electric cylinder.

3. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 2, characterized in that: According to step (1), the specific method is as follows: Solve the inverse kinematic solution by the closed-loop vector method, where the closed-loop vector formula of the mechanism is: q i = P m + b i - a i i = (1,2,3) (1) The inverse kinematic solution of the parallel robot obtained by the closed-loop vector method is expressed as:

4. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 3, characterized in that: According to step (2), the specific method is as follows: Conduct kinematic calibration on the mechanism and establish an error model. First, represent the rotation matrix of the mechanism as: Include all the structural parameters corresponding to the above error parameters into the inverse solution expression to obtain: Where: L = (zc1 - zp + xla1sβ - zla1cαcβ + yla1cβsα), M = (xla1cβ - xc1 + zptβ + zla1cαsβ - yla1sαsβ), N = (q4 + yc1 - yla1cα - zla1sα), O = (zp - zc2 - xla2sβ + zla2cαcβ + yla2cβsα), P = (xla2cβ - xc2 + zptβ + zla2cαsβ + yla2sαsβ), Q = (q5 + yc2 - yla2cα + zla2sα), R = (zp - sβ(xc3 + xla3) + cαcβ(zc3 + zla3) + cβsα(yc3 + yla3)), S = (cβ(xc3 + xla3) - q6 + zptβ + cαsβ(zc3 + zla3) + sαsβ(yc3 + yla3)), T = (cα(yc3 + yla3) - sα(zc3 + zla3)). Further organize the above formula to obtain: Perform total differentiation on both sides of formula (5) to obtain: Where: Solve the system of linear equations of three variables for Equation (6), where are the equation parameters, δα, δβ, δz p are the unknowns, and write the equation in matrix form as: In the formula, let Thus, the mapping relationship between the end pose error and the kinematic error parameters is obtained as: Generally, a tool is installed at the end of the moving platform of a parallel mechanism. Assuming the length of the tool at the end of the moving platform of the mechanism is H, the coordinates of the end point of the tool are expressed in the moving coordinate system o-uvw as: H o =(0 0 H) T , According to the coordinate transformation, its expression in the fixed coordinate system O-xyz is obtained as: Among them, P m is obtained from the self-constraint conditions of the mechanism: P m = (z p tanβ0z p ) T (10) Perform first-order perturbation on both sides of formula (10) to obtain: Δr = (δz p tanβ + z p δαsec 2 β0δz p ) T (11) Convert formula (11) into matrix form to obtain: From the mapping relationship between the skew-symmetric matrix of angular velocity and the rotation matrix Obtain the angular velocity of the moving platform: ΔΩ = (cosβδα δβ - sinβδα) T (13) The relational expression between the position of the tool end point and the pose of the moving platform is: p = r + R p l c (14) Perform first-order perturbation on both sides of formula (14) to obtain: Δp = Δr + ΔΩ × (R p l c ) + R p Δl c (15) where Δp = (Δx p Δy p Δz p ) T represents the position deviation of the end effector in the fixed coordinate system. Δl c =(δd1 δd2 δd3) T refers to the tool error of the end tool in the moving coordinate system; Substitute Equation (15) into Equation (12) to obtain: Further rearrangement gives: Δp i = J i Δε i (i = 1, 2, 3) (17) In the formula: J d The value of is: Thus, the establishment of the error model of this kinematically redundant parallel mechanism is completed.

5. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 4, characterized in that: According to Step (3), its parameter identification steps are as follows: (1)Before starting iterative identification, based on the theoretical structural parameter r k In the workspace of the 2PUPR-PRPU motion-redundant parallel mechanism, select measurement pose data for error parameter identification. Substitute the measurement pose into the inverse solution expression to solve for the corresponding drive values. Considering the current structural parameters in this case, use the forward kinematics of the mechanism to solve for the theoretical pose T of the end effector (k) , and combine with the error model to obtain the error identification matrix under the current structural parameters (2) Let the error parameter of the mechanism be δr k , and the actual parameter of the mechanism is obtained as r k + δr k . Using the forward solution, the end - pose T of the mechanism with structural parameter errors is obtained k . The actual measured pose T is obtained by adding measurement noise m ; (3) Transform T m -T (k) to obtain the pose error δT (k) , and then use the regularized nonlinear least squares method to identify the normalized identification matrix to obtain the error parameters of the mechanism (4) Then judge whether δT (k) -δT (k-1) is less than the identification accuracy index. If this index is not satisfied, update the structural parameters to Repeat the iterative process until the pose error converges, and finally output the identified structural parameters and the structural parameter error.

6. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 5, characterized in that: The parameter identification iteration formula is expressed as: δr k (k + 1) = δr k (k) + ((J k ) T J k + λI) -1 (J k ) T δT k (19) where λ represents the regularization coefficient.

7. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 6, characterized in that: According to Step (4), its specific method is as follows: Before optimizing the measurement points, first pre-select the measurement points. The measurement points should cover the workspace as much as possible and include the measurement points on the boundary of the workspace. After pre-selecting the measurement poses, use the measurement pose optimization algorithm to optimize the poses of the pre-selected measurement points. By introducing the observability index to evaluate the quality of the measurement poses, the observability index is used as an index for pose optimization and is associated with the identification matrix. There are mainly the following types: O3 = σ m (22) where n represents the number of measurement points, and σ1≥σ2≥...≥σ m represents the non-zero singular values of the identification matrix composed of n measurement poses; By comparing the differences of each observability index and its influence on the uncertainty of the end pose, O3 is determined, that is, the smallest non-zero singular value of the identification matrix is the best observability index. Taking O3 as the optimization goal, the pose data set is optimized through the optimization algorithm.

8. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 7, characterized in that: Set the locust optimization algorithm, and its mathematical formula is: X i = S i + G i + A i (26) Among them, X i is the position of the locust, S i is the mutual influence between individuals, G i is the gravity of the locust, A i is the wind force acting on the locust; The social role of locusts has the greatest impact on the position of locusts, and the expression is as follows: where d ij = |x i - x j | is the distance between the i-th locust and the j-th locust; d ij = (x j - x i ) / d ij is a unit vector; s is the calculation method of social force as follows: where f is the attraction intensity, l is the attraction intensity scale, and take f = 0.5, l = 1.

5. Ignoring the effects of gravity and wind, X i can be expressed as:

9. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 8, characterized in that: Taking the locust optimization algorithm as the measurement pose optimization algorithm, its steps for optimizing the measurement pose are as follows: (1) Initialize the parameters, including initial parameters such as the locust population size and the number of algorithm iterations. (2) Each locust represents a pose subset. Calculate the fitness of each locust and the smallest singular value of the identification matrix. (3) Calculate the current best fitness and maximize the smallest singular value. (4) Update the position of the locusts, calculate the fitness after update, compare it with that before update, and judge whether it needs to be updated. (5) Judge whether the number of iterations is reached. If so, the step ends, and the position of the current global optimal solution is the optimal solution.

10. A kinematic calibration and measurement pose optimization method for a motion redundant parallel mechanism according to claim 9, characterized in that: According to Step (5), its specific method is as follows: The driving values corresponding to each driving joint are calculated by inverse-solving the nominal pose at the end, and then the identified structural error parameters and the identified Jacobian matrix at the current pose are substituted to solve the pose error to be corrected In, the kinematic parameters after error compensation are expressed as: Substitute the compensated kinematics into the inverse solution expression for solution to obtain the corrected driving value of the parallel mechanism. Input the above corrected driving value into the control system to complete the compensation of the end pose.

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