Kinematic Error Modeling And Parameter Identifiability Analysis Of Parallel Mechanisms

Kinematic calibration is an effective technology to improve the absolute positioning accuracy of mechanisms.Generally,the kinematic calibration procedure can be divided into four steps:error modeling,configuration measuring,parameter identifying and error compensating.Thereinto,the error model is the fundamental of kinematic calibration,which should satisfy the requirements of completeness,continuity and minimality.Otherwise,it will have a direct impact on computation efficiency,stability and accuracy of parameter identification.Therefore,it is necessary to establish perfect error models for the mechanisms to improve their positioning accuracy to high levels.At present,the researches on the kinematic error modeling of serial mechanism have been relatively perfect.However,for parallel mechanisms,there are still many problems to build the error models that satisfy the requirements of completeness,continuity and minimality.On one hand,in error modeling,the forward kinematics of most parallel mechanisms are difficult to be characterized by analytical form.Therefore,the error models cannot be established by taking the first-order differential of their forward kinematic equations.Currently,the error models of parallel mechanisms are mostly established by taking the first-order differential or performing disturbance analysis of their closed-loop vector equations.However,it is difficult for the closed-loop equations to include all of the kinematic parameters.As a result,the error model cannot meet the requirements of completeness.On the other hand,the closed-loop characteristics of the parallel mechanism cause the idle motion of passive joints,which affects the error transmission relationship and increases the difficulty in analyzing the identifiability of error parameters.At present,the current discussion of this issue is mostly through numerical methods.In addition,the parallel mechanism contains a large number of passive joints,and whether the idle motion of the joints will affect the processes and results of parameter identification also lacks research.In order to overcome these shortcomings,this paper firstly proposes an error modeling method to meet the requirements of completeness and continuity,based on the product of exponentials(POE)formula.On this basis,the method of identifiability analysis for the errors of kinematic parameters is proposed and the approach on how to eliminate the redundant parameters is then given.Besides,in order to enhance the stability and efficiency of the identification algorithm,it is proposed to use the total least squares(TLS)method instead of the general least squares(LS)method to solve the error parameters.Finally,the kinematic calibration experiments on 3-PRRU and6-UPS parallel mechanisms are conducted to validate the effectiveness of the method this paper proposed.The work is divided into the following parts:1)Complete and continuous error models for parallel mechanismsIn order to meet the requirements of complete and continuity,the error modeling method of the limbs in parallel mechanisms is firstly proposed based on POE formula.By using the screw theory,the displacement errors of unmeasurable joints in the limb's error model are eliminated by projecting the constraint twists of the unmeasurable joints'twists onto the pose error of the end effector.Thus the error model of the parallel mechanism can then be obtained by combing all the limbs'error models together.Compared with the conventional error modeling methods through inverse kinematics,this error modeling method involves all potential errors of kinematic parameters and does not need to establish the error model case by case.2)The identifiability analysis of error parameters of parallel mechanismsFirstly,a general method of analyzing the identifiability of error parameters and eliminating the redundant ones is proposed based on the matrix theory.On this basis,the null spaces of the identification matrices in the error models before and after the projection are respectively analyzed.By constructing the standard orthogonal bases of the null and row spaces of the identification matrices,the redundant parameters are theoretically eliminated and the error model that satisfies the requirements of completeness,continuity and minimality can then be obtained.Besides,the equation to determine the maximum number of the identifiable parameters are concluded as N=4r+2p+6.Finally,the error models of cylinder(C),universal(U)and spherical(S)joints are individually analyzed when the structure of these joints are assumed perfect,which provides the theoretical fundamental for the study of the influences of these joints'structure errors on calibration results.3)The analysis of the stability of the identification algorithmIn order to ensure a stable process of parameter identification,the disturbance sources of the error model?e(28)J_p?p are analyzed,and the conclusion is that both the identification matrixJ_p and the pose error?e exist disturbances.The least-squares method can obtain the ideal result when only the pose error?e exists disturbances.If the identification matrix has disturbances simultaneously,the calculation error will increase.Therefore,the total least squares method is used instead of the least squares method to identify the error parameters.This method has better numerical stability than least-squares method when solving the problem of both the identification matrix and pose error exist disturbances.Finally,numerical simulations are conducted on the 3-PRRU,3-PRS,6-UPS parallel mechanisms and so on,the results of the cases show that the total least squares method has better stability than the squares method,and better calibration results have been obtained by TLS method.4)Experimental verification by kinematic calibrations of parallel manipulatorsTo validate the effectiveness of the method this paper proposed,the 3-PRRU and6-UPS parallel prototypes are designed and established.The kinematic calibrations for both the manipulators are conducted and the influences of the manufacture errors in U joints on calibration result are investigated through the calibration of 3-PRRU manipulator.The effectiveness of the TLS method is verified and the advantage of computational stability is shown by comparing with the LS method.The calibration results of both the manipulators are compared with the ones obtained from the error models which are established by conventional method.The comparing results show that the absolute positioning accuracy will be higher when the kinematic calibrations are conducted through the error models that are established based on the method in this paper.Error models satisfying the requirements of completeness,continuity and minimality are established for parallel mechanisms.An analytical method of determining and eliminating redundant error parameters is proposed.The influences of idle motions of passive joints on parameters identifiability and identification stability are revealed.Finally,the researches in this dissertation form a theoretical basis for the kinematic calibration of parallel mechanisms,make a theoretical basis for efficient,stable,and high-precision parameter identification,and provide technical assurance for the high-precision operation of parallel mechanisms.