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Kinematic calibration of a 3-PRRU parallel manipulator based on the complete, minimal and continuous error model

https://doi.org/10.1016/j.rcim.2021.102158Get rights and content

Highlights

  • The CMC error model of 3-PRRU parallel manipulator is established by POE formula.

  • The maximum number of identifiable parameters is 60, which accords with 4r+2p+6.

  • The manipulator possess a much better calibration result than conventional method.

  • The method this paper proposed can be applied to most parallel manipulators.

Abstract

Kinematic error model plays an important role in the kinematic calibration of robot manipulators. In order to guarantee the best calibration result, the error model should meet the requirements of completeness, minimality and continuity. For the conventional serial robots, these issues have been intensively studied and a consensus has been reached during the past decades. However, for parallel manipulators, the problem of minimality, namely the determination of identifiable parameters mainly relies on numerical studies. There is a lack of effective methods to analyze the parameters’ identifiability in the error modeling process. Besides, there is still no agreement on the formula of the maximum number of identifiable parameters. In our previous work, a complete, minimal and continuous error modeling approach for parallel manipulators has been proposed based on product of exponential (POE) formula and the equation that the maximal number of identifiable parameters 4r+2p+6 was proved. However, the correctness and effectiveness of this method in kinematic calibration have not been verified on any parallel manipulator or robot. Therefore, in this paper a kinematic calibration is conducted on a prototype of a 3-PRRU parallel manipulator based on the proposed error model. The method of validating the completeness and minimality are respectively given by numerical simulations. Furthermore, to illustrate the advantage of the proposed error model, the kinematic parameters of the prototype are also calibrated based on the conventional error model which is derived by means of differentiating the limbs’ inverse kinematics. By comparing the calibration results, it can be concluded that the error model satisfying the three criteria can better improve the positioning accuracy. In addition, since the calibration method is described uniformly, it can be applied to most parallel manipulators by referring to the calibration procedure this paper presented.

Introduction

Due to manufacturing tolerances, installation errors and link offsets, the actual kinematic parameters of a robot manipulator will not exactly match their nominal values. As a result, the pose of the end-effector or moving platform will have a deviation from the desired one. To overcome this shortcoming in a effective and economical way, kinematic calibration attracts intensively attentions in the past several decades [1], [2]. Generally, calibration methods are usually based on kinematic error models [3] or intelligent algorithms [4]. Compared with the latter methods, such as the particle swarm optimization [5], neural network [6] or whale swarm algorithm [7], the former one can reveal the error transfer rules between kinematic and pose errors, and usually has higher computational efficiency. Therefore, most kinematic calibrations are based on error models and the research can be mainly classified into four categories: error modeling, pose measuring, parameter identifying and error compensating [8]. Among these, kinematic error model plays an important role in both iterative computation stability and calibration result. At present, a consensus has been reached that the error model for kinematic calibration should satisfy the criteria of completeness, continuity and minimality [9], [10].

During the past decades, the issues of kinematic calibration have been intensively studied for the conventional serial robots. Denavit–Hartenberg (D–H) based approaches [1], [2], [11], [12], the continuous and parametrically complete (CPC) model [13], [14], the single joint method [15] and the models based on POE formula [16], [17], [18] were successively presented. Compared with other error modeling methods, the error models based on the POE formula have inherent advantages because of the smooth property of the exponential map from the Lie algebra se(3) to the Lie group SE(3). First, the kinematic parameters in error models based on the POE formula vary smoothly with changes in joint axes, which guarantees that the error models are singularity free. Second, because any rigid-body motion can be regarded as a screw motion corresponds to a certain twist, these error models can also satisfy the completeness requirement [19]. In addition, since all joint axes are described based on the line geometry in POE formulation, it can be uniformly applied to the manipulators both with revolute and prismatic joints [17]. Because of these advantages, most of the discussions on completeness, continuous and minimality of the kinematic error model, especially the issue of minimality, are based on the POE formula. At present, there are several methods [20], [21], [22], [23], [24] based on POE formula can be applied to establish the complete, continuous and minimal error models for the kinematic calibration of serial robots. And a consensus has been reached that the maximum number of identifiable kinematic parameters for serial robots is 4R+2P+6, where R and P represent the numbers of revolute and prismatic joints, respectively.

Due to complex topologies of parallel manipulators, the calibration models are mostly established by inverse kinematics or closed-form features [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. However, to simplify the procedures of kinematic analysis and error modeling, some structures or links have to be assumed perfectly manufactured and assembled. On one hand, these assumptions make it difficult to ensure the error models meet the complete requirement, which limits the accuracy improvement of kinematic calibration. On the other hand, the ignored parameters may make troubles for accuracy analysis or tolerance design because the influences of these error parameters on positioning accuracy cannot be obtained. To overcome these shortcomings, the error models established through limbs are proposed such that all of the potential error sources can be involved [38]. Since the topologies of limbs are similar with serial robots, most of the error modeling methods proposed for serial chains can be used as basics for parallel manipulators. For example, in order to evaluate the accuracy of a 6-UPS parallel manipulator, Wang and Masory [39], [40] developed a kinematic model by D–H method to accommodate all of the potential kinematic errors. Liu [41] proposed an error modeling approach by screw theory, based on which the parameters of a 3-DOF parallel manipulator are identified [42]. In order to investigate the influence of joint backlash on positioning accuracy, Meng [43] established an error model based on POE formula. Furthermore, by taking use of the finite and instantaneous screw theory, and the reciprocal twist and wrench, a generic error modeling method for both serial and parallel robots was presented in [44]. As another merit of this kind of method, the continuous requirement is no longer a critical problem because this issue in serial robots has been solved. Therefore, the minimal requirement, namely the identifiability condition for the error parameters, becomes the main issue for the reliable and stable identification of parallel manipulators’ kinematic parameters.

So far, the identifiability of kinematic parameters are mostly analyzed by numerical methods or experience, which lacks mathematical proof. For example, Besnard and Khalil [45] investigated the identifiability of kinematic parameters for a 6-UPS parallel manipulator based on QR decomposition. Joubair [46] determined the redundant parameters by an iterative algorithm for a hybrid 6-DOF medical robot. In order to eliminate the unidentifiable parameters, correlation analysis was conducted on the column of identification matrix of a 5-DOF hybrid machine tool by Tian [47]. Furthermore, another question about parameter identifiability is how to get the maximum number of identifiable parameters for a specific parallel manipulator. Due to the limited study of parameter identifiable analysis, the equations about the maximum identifiable parameters are also based on experience. In 1989, Lin [48] firstly gave an equation C=3R+P+SS+E+6+6 for single-loop parallel robots with R revolute joints, P prismatic joints, SS pairs of S-joints and E encoders. Later, Vischer [49] extended this conclusion for multi-loop parallel robots by adding the number of loops F and the arbitrarily located frames L into the equation which was given as C=3R+P+SS+E+6L+6(F1). However, Vischer did not give a proof of this equation and states that “this equation was empirically tested on several examples and seems to be valid...”. In 2007, Legnani [50] proposed a more general equation C=3R+P+2C+SI+E+6L+6(F1) by considering the number of cylindrical joints C and separating SS joints into SS links (SI=+1) and SPS legs (SI=1), but this equation is also experience based and lacks a mathematical proof. Recently, Yin [51] proposed a screw based approach to determine the identifiable parameters, and the maximum number of independent kinematic parameters is concluded as N=Σi=1l(4ri+2pi+6Np,iN0,i), where l denotes the number of limbs, Np,i and N0,i represent the numbers of passive joints and unidentifiable structural errors detected by screw theory in limb i, respectively. However, taking 6-UPS parallel manipulator as an example, the result derived by Yin’s equation has a bit difference from the numerical result [39] and empirical formula [52]. Until now, there is still no agreement on the formula of the maximum number of identifiable parameters.

In our previous work [53], a complete, minimal and continuous error model for parallel manipulators has been proposed based on POE formula and the equation of the maximal number of identifiable parameters 4r+2p+6 was theoretical proved. However, the conclusion and the effectiveness of this method has not been verified on any parallel manipulator, and the applicability of this approach still needs to be validated. Thus in this paper, the kinematic parameters of a 3-PRRU parallel manipulator is calibrated based on the proposed method. By numerical simulations and experiments, the correctness and applicability of the error model are validated. Furthermore, to illustrate the effectiveness and advantage, the calibration result is also compared with the one that the conventional error model obtains.

This paper is organized as follows. In Section 2, the 3-PRRU parallel manipulator is introduced and inverse kinematics is briefly given. In Section 3, the error model satisfying the requirements of completeness, minimality and continuity is established. Numerical simulations and experiments are then carried out in Section 4 to verify the correctness and effectiveness of the error model. Finally, some conclusions are drawn in Section 5.

Section snippets

Architecture and kinematics of the studied 3-PRRU parallel manipulator

In this section, the studied 3-DOF 3-PRRU parallel manipulator is firstly introduced, where P denotes the actuated prismatic joint. As mentioned above, in order to make the error model meet the requirement of completeness, the PRRU limbs should be regarded as PRRRR serial chains. However, it will become more difficult to solve the forward and inverse kinematics of the parallel manipulator, especially for the case that the kinematic parameters are no longer nominal values. In order to deal with

Complete, minimal and continuous error model of the parallel manipulator

In this section, the kinematic error model of the parallel manipulator that contains all error parameters will be firstly built up by local POE formula. The identifiable analysis of kinematic parameters are then given based on null-space characteristics of the error transformation matrices in two steps.

Numerical simulations and experimental validation

In the previous section, a complete, minimal and continuous error model of the 3-PRRU parallel manipulator has been established, but the correctness and applicability still need to be validated. Therefore, in this section, the completeness, minimality and the correctness of the error model on kinematic calibration are verified by numerical simulations. Furthermore, in order to illustrate the effectiveness and advantage of the proposed method, a conventional error model which is derived by means

Conclusion

This paper presents a feasible and effective kinematic calibration method for 3-PRRU parallel manipulator based on the complete, minimal and continuous error model. The completeness and minimality are verified based on the numerical simulations, respectively, and the number of identifiable parameters is accord with our previous conclusion 4r+2p+6. After the experiment of kinematic calibration, the mean position and orientation errors are reduced from 4.60×103m and 1.18×102rad to 1.93×105m

CRediT authorship contribution statement

Lingyu Kong: Conceptualization, Methodology, Software, Investigation, Writing - original draft. Genliang Chen: Conceptualization, Methodology, Formal analysis, Supervision. Hao Wang: Conceptualization, Supervision. Guanyu Huang: Investigation, Writing - original draft. Dan Zhang: Supervision, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported in part by the Natural Science Foundation of Zhejiang Province of China (Grant No. LQ20E050008 and LQ21F030003), and the Leading Innovation and Entrepreneurship Team of Zhejiang Province of China (Grant No. 2018R01006).

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