A minimal-error-model based two-step kinematic calibration methodology for redundantly actuated parallel manipulators: An application to a 3-DOF spindle head

https://doi.org/10.1016/j.mechmachtheory.2021.104532Get rights and content

Highlights

  • l

    A set of principles of eliminating redundant geometric source errors in RAPMs is proposed.

  • l

    An kinematic error model including least number of geometric source errors is established for RAPMs.

  • l

    A hierarchical-error-identification based two-step kinematic calibration methodology is proposed for RAPMs.

Abstract

Zero offsets and geometric source errors will significantly degrade the kinematic accuracy of redundantly actuated parallel manipulators (RAPMs). To relieve the influences of these factors, this paper presents a minimal-error-model based two-step kinematic calibration methodology for this type of parallel manipulators. A novel 3-DOF spindle head with a 2UPR&2RPS topology is taken as an example to demonstrate the kinematic calibration methodology. The proposed kinematic calibration methodology includes three critical steps: (1) a set of general principles is proposed to eliminate redundant geometric source errors in the manipulator to derive a minimal error model that includes the least number of geometric source errors; (2) a sensitivity analysis is carried out using the Monte-Carlo simulation to reveal the relative impact of geometric source errors on the terminal accuracy; (3) a hierarchical identification strategy composed of a coarse identification and a fine identification is proposed, based on which a two-step calibration methodology is constructed. Finally, a set of calibration experiments is performed to verify the effectiveness of the proposed calibration methodology.

Introduction

Integrating a three degree-of-freedom (3-DOF) parallel manipulator as a spindle head with a 2-DOF serial module to construct a hybrid five-axis machining unit is regarded as a promising solution for machining large structural components with complex geometrical surfaces [1], [2], [3]. A typical commercially successful application of this proposition can be traced back to the Eco-speed machining center which combines a parallel spindle head named Sprint Z3 with two orthogonal gantries [4]. Similar attempts and applications of serial-parallel hybrid 5-axis machining units can be traced in the references [5], [6], [7]. Since the 2-DOF serial sliding gantry has been well-developed with high precision, the kinematic accuracy of such kind of hybrid five-axis machining unit is highly dependent on the kinematic accuracy of its parallel functional module, i.e., the 3-DOF spindle head. As a result, it is of great importance to realize high kinematic accuracy of the parallel manipulator before it is functioned as a spindle head for high quality machining.

According to the relationship between the number of DOFs and that of actuators, parallel manipulators can be roughly classified into redundantly actuated ones and non-redundantly actuated ones. Compared with their counterparts of non-redundantly actuated forms, redundantly actuated parallel manipulators (RAPMs) often claim comparative advantages of better dexterity, enhanced stiffness and better dynamic performances [8,9]. Therefore, they have attracted intensive attentions from both academic and industrial societies in recent years [10], [11], [12]. When a 3-DOF RAPM is used as spindle head to perform high quality machining, its kinematic accuracy must be guaranteed. To realize a satisfactory kinematic accuracy, an economic yet efficient way of kinematic calibration needs to be carried out, which usually includes the procedures of error modeling, error measurement, error identification and error compensation [13,14]. Among these four procedures, error modeling and error identification are the most important ones. To be specific, error modeling is the theoretical basis for kinematic calibration, which reveals the propagation mechanism between the geometric source errors and the terminal errors. On the basis of error modeling, error identification aims to clarify geometric source errors with measured terminal errors, providing detailed parameters for error compensation.

For the first critical issue of error modeling, an error model is generally acquired by the first-order perturbation of the ideal kinematic equations of a parallel manipulator, where Denavit–Hartenberg (D–H) method, vector method and screw method are commonly used. Following this track, He [15] established an error model for a hybrid manipulator with modified D–H method; Sun [16] formulated an error model for a 3-DOF parallel manipulator by using the vector method; and Frisoli [17] used the screw theory to construct an error model of a 3URU spatial parallel manipulator. Besides these commonly used methods, Chen [18] proposed an error modeling approach based on Product-Of-Exponential (POE) formula for parallel manipulators. It is worthy pointing out that due to the introduced redundant actuations and/or constraints, error mapping relationships are altered while nonnegligible non-linear structural deformations might be generated in RAPMs and over-constrained parallel manipulators [19], [20], [21]. Realizing this point, Jiang [22] established an error model which considers the link deformations caused by constraint forces and inertial loads for a redundant planar parallel kinematic machine. Li [23] derived an error model for an over-constrained parallel kinematic machine with the consideration of force-induced deformations calculated by energy method. To avoid the complex calculation of non-structural deformations, Jeon [24] projected the constraint force terms onto the orthogonal complementary terms during the error modeling of RAPMs. By using similar projection technique, Luo [25] avoided measuring the stiffness of active joints and developed an error model for a 3-DOF RAPM. By tracing these previous investigations, one may find that after considering redundant actuations and constraints the error model of an RAPM often become very complicated. This, in turn, will bring considerable difficulties to the realization of accurate error identification. In parallel manipulators, there are some geometric source errors which are dependent on the other geometric source errors thus can not independently affect the terminal errors of a parallel manipulator, which are called redundant geometric source errors [26]. Seen from the previous study [27], the existence of redundant geometric source errors may introduce linear independence in the error Jacobian matrix, then the loss of identifiability is generated. Therefore, it's necessary to eliminate redundant geometric source errors during the error modeling process in order to simplify the calculation and improve the feasibility of error identification.

For error identification, the least square method is most frequently adopted. For example, Huang [28] used the least square method to identify the real parameters of a 6-DOF parallel kinematic machine. Yang [29] employed the least square method to recognize the structural parameters of parallel mechanisms with 6-DOFs. To achieve better identification accuracy, some optimization methods have been proposed and applied. For instance, Rosyid [30] and Li [31] adopted an iterative method for error identification of a hybrid kinematic machine and a 3-P(Pa)S parallel spindle head, respectively; Wu [32] solved the kinematic parameters in symmetrical subspaces of the workspace and then adopted the arithmetical mean values as identification results when calibrating parallel kinematic machines; Hu [33] separated the whole set of error parameters into several subsets and then identified the error subsets sequentially for a 6-DOF parallel manipulator; Russo [34] identified the geometric parameters of a reconfigurable Gough-Stewart manipulator with no priori knowledge of the location of any passive joint by expanding normal identification models.

When using the least square method for error identification of RAPMs, the key point is how to construct the identification matrix. However, in real practice, the multi-collinearity problem among measured configurations, the ill-conditioned poses and the measurement noises would all weaken the robustness of the identification matrix, causing non-negligible identification deviations. To deal with the multi-collinearity problem, Huang [35] employed the Liu estimator to perform the error identification for a 6-DOF hybrid robot. To solve the ill-posed problem, Kong [36] adopted the regularization method for error identification of a 3-DOF parallel manipulator. To improve the robustness of the identification matrix, Bai [37] optimized the measurement pose selection to reduce the condition number of the identification matrix to a given threshold. Tian [38] conducted a statistical analysis to compare the identification effectiveness of four regularization identification algorithms on a 5-DOF hybrid kinematic machine. Rather than making optimization based on the least square method, Sun [39] formulated an objective function of identification and adopted a hybrid genetic algorithm to identify the errors in a 2-DOF over-constrained parallel mechanism.

Besides geometric source errors induced by imperfect manufacturing and assembly of the parts, source errors in a parallel manipulator also include zero offsets induced by home position errors and actuator encoder offsets. Under practical operations of parallel manipulators, zero offsets affect the terminal accuracy as well as geometric source errors. In view of this, Yao [40] and Qian [41] considered zero offsets during error modeling of parallel manipulators, then identified zero offsets and geometric source errors simultaneously. Regarding position errors of parallel robots are mainly caused by the zero errors, Mei [42] established a position error model of a 2-DOF parallel robot containing zero errors and geometric errors, then identified zero errors only. It has to be noted that, zero offsets and geometric source errors are generated by different reasons, usually having different orders of magnitudes thus exerting different impacts on the terminal accuracy of parallel manipulators. Therefore, it's necessary to take both zero offsets and geometric source errors into an error model and then calibrate them separately.

The present study proposes a minimal-error-model based two-step kinematic calibration methodology for RAPMs, bearing with the thoughts of: (1) establishing a simplified error model by eliminating redundant geometric source errors; (2) identifying and calibrating geometric source errors and zero offsets separately. For this purpose, a newly proposed 3-DOF spindle head is taken as an example to demonstrate the establishment of error model and the methodology of kinematic calibration. The rest of the paper is organized as follows: Section 2 formulates the kinematic equations after a brief structure description for the proposed 3-DOF spindle head. Then a set of elimination principles is proposed to reduce the number of geometric source errors in the spindle head, based on which a minimal error model is established. Section 3 performs a sensitivity analysis by using Monte-Carlo simulation to reveal the relative influences of geometric source errors on the terminal accuracy. In Section 4, a two-step calibration methodology is proposed on the basis of a hierarchical identification strategy. A set of calibration experiment is carried out to verify the effectiveness of the proposed calibration methodology in Section 5. Finally, some conclusions are drawn.

Section snippets

Structure description and kinematic formulation

In this section, a newly proposed 3-DOF spindle head is taken as an example to demonstrate the error modeling process. The virtual prototype and schematic diagram of the spindle head are shown in Fig. 1.

As can be seen from Fig. 1, the topological architecture behind the 3-DOF spindle head is a RAPM of 2UPR&2RPS. Herein, `U', `R', `S' and `P' represent universal joint, revolute joint, spherical joint and actuated prismatic joint, respectively. It consists of a fixed base, a moving platform, two

Sensitivity analysis

In this section, a sensitivity analysis is conducted to sort out geometric source errors that have relatively `strong' and `weak' impact on the accuracy of the moving platform. Inspired by previous studies, a sensitivity index can be formulated through the error coefficient matrix. For the geometric source error δn at the nth row of matrix ɛ, the sensitivity coefficients of terminal position and orientation accuracy are established as [45]{μrn=i=13(Jεa)i,n2μθn=i=13(Jεb)i,n2where (Jεa)i,n

Two-step calibration methodology

In the present work, the selected error measurement device a Leica AT960 laser tracker. As an external distance measurement device, a laser tracker measures the position error of a laser reflector point Om instead of the central point O of the moving platform. As shown in Fig. 6, the laser reflector point, i.e. the measurement point, Om is located in the w axis of the body-fixed frame Ouvw of the moving platform.

The position error model of the measurement point Om can be expressed asem=Δr+(J3

Experimental validation

An experimental setup for both coarse and fine calibration is developed and shown in Fig. 8. The 3-DOF spindle head is driven by four servo motors, which are controlled by the self-developed NC system. By importing the motion data into the NC system, the moving platform can be adjusted to a target posture. Based on this, a Leica AT960 laser tracker is used to conduct the calibration experiment which tracks the position of the laser reflector point.

In order to achieve satisfactory accuracy and

Conclusions

This paper proposes a minimal-error-model based two-step kinematic calibration methodology for RAPMs. Based on the conducted investigations, some conclusions are drawn as follows.

  • (1)

    Based on the limb structure properties and correlations between delicately defined joint frames, a set of general principles is proposed to eliminate redundant geometric source errors in the RAPMs.

  • (2)

    A minimal error model with the least number of geometric source errors is established to derive an analytical error mapping

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 work is supported by the Open Fund of the State Key Laboratory for Mechanical Transmissions, Chongqing University (Grant no. SKLMT-ZDKFKT-202003), the Industry-Academy Cooperation Project of Fujian Province (Grant no. 2019H6006) and the Natural Science Foundation for Distinguished Young Scholar of Fujian Province (Grant no. 2020J06010). The corresponding author is also thankful to the sponsor of National Natural Science Foundation of China (Grant no. 51875105). The authors would like to

References (47)

  • J. Zhang et al.

    Geometric error analysis of an over-constrained parallel tracking mechanism using the screw theory

    Chin. J. Aeronaut.

    (2019)
  • W. Khalil et al.

    Modeling, Identification and Control of Robots

    (2002)
  • T. Huang et al.

    A general and novel approach for parameter identification of 6-DOF parallel kinematic machines

    Mech. Mach. Theory

    (2005)
  • L. Yang et al.

    Numerical simulation of calibration algorithm based on inverse kinematics of the parallel mechanism

    Optik

    (2019)
  • T. Li et al.

    Kinematic calibration of a 3-P(Pa)S parallel-type spindle head considering the thermal error

    Mechatron

    (2017)
  • J. Wu et al.

    A systematic optimization approach for the calibration of parallel kinematics machine tools by a laser tracker

    Int. J. Mach. Tools Manuf.

    (2014)
  • Y. Hu et al.

    Kinematic calibration of a 6-DOF parallel manipulator based on identifiable parameters separation (IPS)

    Mech. Mach. Theory

    (2018)
  • M. Russo et al.

    A calibration procedure for reconfigurable Gough–Stewart manipulators

    Mech. Mach. Theory

    (2020)
  • T. Huang et al.

    Kinematic calibration of a 6-DOF hybrid robot by considering multicollinearity in the identification Jacobian

    Mech. Mach. Theory

    (2019)
  • L. Kong et al.

    Kinematic calibration and investigation of the influence of universal joint errors on accuracy improvement for a 3-DOF parallel manipulator

    Robot. Comput. Integr. Manuf.

    (2018)
  • W. Tian et al.

    Kinematic calibration of a 5-DOF hybrid kinematic machine tool by considering the ill-posed identification problem using regularisation method

    Robot. Comput. Integr. Manuf.

    (2019)
  • J. Dai et al.

    Interrelationship between screw systems and corresponding reciprocal systems and applications

    Mech. Mach. Theory

    (2001)
  • Y. Song et al.

    Kinematic calibration of a 5-DOF parallel kinematic machine

    Int. J. Precis. Eng. Manuf.

    (2016)
  • Cited by (16)

    • A novel hybrid machine tool integrating a symmetrical redundantly actuated parallel mechanism: Design, kinematics, prototype and experiment

      2022, Mechanism and Machine Theory
      Citation Excerpt :

      In view of these merits, a number of 1T2R RAPM were proposed and applied for constructing hybrid machine tools [28–30]. From the perspective of mechanism, a 1T2R RAPM can be constructed through the followings three traditional manners: (1) replacing one or more passive joints with active joints in an original parallel mechanism [30]; (2) adding a full-mobility active kinematic chain into an original parallel mechanism [29,31]; (3) introducing a lower-mobility active kinematic chain to an original parallel mechanism [32,33]. By adopting above three manners, a series of 1T2R RAPMs have been proposed in the past years [34–36].

    View all citing articles on Scopus
    View full text