Natural frequency analysis of parallel manipulators using global independent generalized displacement coordinates

doi:10.1016/j.mechmachtheory.2020.104145

Mechanism and Machine Theory

Volume 156, February 2021, 104145

https://doi.org/10.1016/j.mechmachtheory.2020.104145 Get rights and content

Highlights

•
A method for extracting the global IGDC is proposed.
•
The elastodynamic model is proposed based on the global IGDC.
•
A method for rapidly evaluating the fundamental frequency of the PMs is proposed.

Abstract

A modeling approach is proposed for analyzing the natural frequencies of parallel manipulators (PMs) using the global independent generalized displacement coordinates (IGDC). To avoid the constraint equations while solving the dynamic equation, all of the compatibility conditions must be integrated into the overall stiffness and mass matrices. The global IGDC is proposed to solve this problem. We establish one set of nonsingular independent displacement coordinates of the joint connection points using a multipoint constraint element and singularity assessment of the mapping matrix in the workspace. The global IGDC is obtained by combining the coordinates of the joint connection points and the inner nodes of the components. Matrix structural analysis (MSA) is used in combination with the global IGDC to establish the model for analyzing the natural frequencies of the PMs. The 2UPR-RPU and 2UR-2RPU PMs are presented to illustrate the effectiveness of the proposed model. We investigate the influence of the number of element divisions on the natural frequencies considering the member as a single element to ensure accurate and efficient computation of the fundamental frequency of the PMs.

Introduction

Parallel manipulators (PMs) have become important in developing high-speed machining owing to their excellent dynamic performance [1–2] as demonstrated by the commercial success of the Sprint Z3 and Tricept robots [3], [4]–5]. The set of low-order natural frequencies is an important index for the dynamic performance of the mechanisms. Higher natural frequency, particularly the fundamental frequency, means higher control bandwidth, which can reduce the mechanisms’ vibration response. Machine vibration affects the processing accuracy, reduces the life of the cutter, and even causes resonance damage to the machine [6]. Therefore, investigating the dynamic performance of PMs and understanding their natural frequency distributions in the workspace are very important for controlling machine vibration.

The natural frequency is an inherent property of a mechanism and closely related to the mechanism's structural parameters and configuration. Before calculating the natural frequencies, it is necessary to establish the overall stiffness and mass matrices of the mechanism, which should be established under the same coordinate system [7], [8]–9]. The key to this is to comprehensively consider the compatibility conditions of the mechanism and extract its global independent generalized displacement coordinates (IGDC) under the global coordinate frame. The natural frequency analysis of PMs has been intensively investigated over the past decades. The approaches toward establishing the dynamic model can be approximately classified into four categories: finite element method (FEM), virtual joint method (VJM), assumed modes method (AMM), and matrix structural analysis (MSA).

Bouzgarrou [10] established the static and dynamic model of a 4-DOF PM using the FEM. Son [11] used FEM to analyze the dynamic performance of a 3-DOF reconfigurable PM. Palmieri et al. [12] analyzed the natural frequencies of a purely moving PM in the workspace by combining the FEM and a polynomial regression method. Ma et al. [13] analyzed the distributions of the natural frequencies of a PM in the workspace by combining computer-aided design (CAD) and computer-aided engineering (CAE). Through the batch method, the mechanism can automatically update the finite element model as the mechanism's configuration changes. The links and joints can retain their actual shape and size in the FEM, and thus accurate results can be obtained [14], which is desirable by engineers. The finite element model requires re-meshing for recalculation as the configuration of the mechanism changes, which is very time-consuming. Therefore, the FEM is more suitable for use in the final design stage or for verifying the validity of other analytical models.

The VJM considers the link as a pseudo-rigid body supported by a spring with six degrees-of-freedom (DOFs) at each end to describe its inertia and flexibility [15], [16]–17]. Vu and Kuo [18] used the VJM to establish the dynamic equations of the planar serial mechanism under the action of gravity and external loading. Mei and Zhao [19] established the dynamic model of the 6-RSS PM based on the VJM. The MSA or the substructure synthesis technique is based on the main concept of the FEM and divides the component into a certain number of elements, which can achieve a good balance between the calculation time and the calculation accuracy [20], [21]–22]. Pham et al. [23] established the dynamic model of a planar flexible PM based on the MSA, and analyzed the distribution of the first-order natural frequency in the workspace. Zhang and Zhao [24, 25] established the dynamic model of the 3-PRS PM based on the MSA. Lian et al. [26–27] used the semi-analytical method to obtain the mass and stiffness matrices of the components, and combined this with substructure synthesis technology to obtain the dynamic control equation of the mechanism. Liu et al. [28–29] established the dynamic equations of the 8PSS and TriMule robots based on substructure synthesis technology, and obtained the distribution of the natural frequencies in the workspace. However, the above-mentioned studies have not established a general method for extracting the global IGDC in the global coordinate system. The AMM considers the rod as a continuum beam, and assumes that the deformation of the rod is a linear superposition of its first few modes. Zhang et al. [30], [31], [32], [33]–34] established the dynamic equations of the planar PM based on the AMM. Notably, the AMM can be used to analyze the problems of the forced vibration of damped and non-conservative systems, but it is difficult to assume accurate modal shapes for complex mechanisms. Additionally, the above-mentioned studies have not applied the AMM to spatial PMs.

The main contribution of this study is twofold. First, we propose a method for extracting the global IGDC by combining the multipoint constraint element and singularity assessment criterion of the mapping matrix, and an elastodynamic model of the PMs based on the global IGDC. The proposed model contains all of the required body-to-body and body-to-ground constraint conditions, and has a clear physical meaning. Second, using the proposed model, we further propose a method for rapidly evaluating the fundamental frequency of the PMs by investigating the influence of dividing the rods into different numbers of elements on the natural frequencies of the PMs. The proposed method can simplify the modeling procedure used for natural frequency analysis and greatly improve the efficiency of calculating the fundamental frequency.

The structure of this paper is organized as follows. In Section 2, the process for the natural frequency analysis of the PMs using MSA and global IGDC is presented. In Section 3, the natural frequencies of the 2UPR-RPU PM are investigated using the proposed method, and the influence of the number of divided elements on the natural frequencies is analyzed. In Section 4, the natural frequencies of a redundantly actuated 2UPR-2RPU PM are compared with those of its non-redundantly actuated counterpart, namely, 2UPR-RPU PM. In Section 5, the conclusions drawn from this study are presented.

Section snippets

Natural frequency analysis of parallel manipulators

Without loss of generality, let us consider the PM shown in Fig. 1, which consists of a moving platform connected to a base by n limbs. The coordinate frames O-XYZ and o-xyz are attached to the fixed base and moving platform, respectively; o_ij-x_ijy_ijz_ij is the element coordinate frame of the j^th rod in the i^th limb.

In this study, the following assumptions were made with regard to the PMs:

(i)
The fixed base, moving platform, actuators, and joints are rigid.
(ii)
The friction between the components is

Example 1: 2UPR-RPU PM

As we know, the deformation compatibility relationships of the sphere joints are easy to achieve, in order to illustrate the versatility of the proposed modeling, the 2UPR-RPU overconstrained PM developed in our laboratory is selected as the example to verify the effectiveness of the modeling. The schematic diagram of the 2UPR-RPU PM [36–37] is shown in Fig. 4. The moving platform is connected to the fixed platform by two UPR limbs and one RPU limb. The UPR limbs connect to the fixed platform

Example 2: 2UPR-2RPU PM

To further verify the proposed model and compare the dynamic performance of the non-redundantly and redundantly actuated PMs, a redundantly actuated 2UPR-2RPU overconstrained PM is considered as another example. Fig. 7 shows the diagram obtained by adding an actuated RPU limb to the 2UPR-RPU overconstrained PM. According to the analysis process presented in Section 3, the global IGDC can be obtained as follows: $U = {[\begin{matrix} φ_{B 1 x} & φ_{B 1 y} u_{i n, 1} & φ_{A 1 y} & φ_{B 2 x} & φ_{B 2 y} u_{i n, 2} & φ_{A 2 y} & φ_{B_{3} x} & u_{i n, 3} & φ_{A 3 x} & φ_{A 3 y} φ_{B_{4} x} u_{i n, 4} φ_{A 4 x} φ_{A 4 x} & Δ_{p}^{T} & φ_{p}^{T} \end{matrix}]}^{T} .$

The

Conclusions

This paper describes the use of MSA and global IGDCs to obtain the overall mass and stiffness matrices of the mechanisms, and thereby obtain the natural frequencies. In this study, the moving platform and joints were assumed to be rigid, and the spatial elastic deformation of the rods was considered. A key feature of this model is that the global IGDCs, which was established using a multipoint constraint element, linear algebra, and the singularity assessment criterion of the mapping matrix,

Declaration of Competing Interest

The authors declare that there is no conflict of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (NSFC) (grant numbers 51775513 and U1713202), the Natural Science Foundation of Zhejiang Province (LY17E050028), and the Public Welfare Research Project of Jiaxing city, Zhejiang province, China (Grant no. 2020AY10013).

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Natural frequency analysis of parallel manipulators using global independent generalized displacement coordinates

Highlights

Abstract

Introduction

Section snippets

Natural frequency analysis of parallel manipulators

Example 1: 2UPR-RPU PM

Example 2: 2UPR-2RPU PM

Conclusions

Declaration of Competing Interest

Acknowledgments

Cirp. Ann. Manuf. Techn.

Mech. Mach. Theory.

Robot. Auton. Syst.

Mech. Mach. Theory.

Int. J. Mach. Tool. Manu

Robot. Cim. -Int Manuf.

Mech. Mach. Theory

J. Mech. Robot.

Precis. Eng.

Robot. Cim. – Int. Manuf.

Mech. Mach. Theory.

Robot. Cim. – Int. Manuf.

J. Sound Vib.

Mech. Mach. Theory.

Application of PKM in aerospace manufacturing-high performance machining centers ECOSPEED, ECOSPEED-F and ECOLINER

The tricept robot: dynamics and impedance control

IEEE-ASME T. Mech.

Cartesian workspace optimization of Tricept parallel manipulator with machining application

Robotica

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J. Comput. Nonlin. Dyn.