Research paper
Stiffness modeling and analysis of a novel 5-DOF hybrid robot

https://doi.org/10.1016/j.mechmachtheory.2017.12.009Get rights and content

Highlights

  • Semi-analytical stiffness model is formulated by treating the planar linkage within the robot as a compound joint.

  • The Cartesian stiffness matrix is explicitly expressed in terms of the compliance matrices down to the joint and link level.

  • Stiffness calculated by the approach matches very well with that obtained by FEA in terms of magnitude and distribution.

Abstract

This paper deals with the stiffness modeling and analysis of a novel 5-DOF hybrid robot named TriMule which is composed of a 3-DOF positioning parallel mechanism plus a 2-DOF wrist. The robot is especially designed as a compact yet rigid module suitable for large part on-site machining. Mainly drawing on screw theory, a semi-analytic stiffness model of the robot is formulated by taking into account the component compliances associated with the elements of both the parallel mechanism and the wrist, resulting in the Cartesian stiffness matrix that can explicitly be expressed in terms of the compliance matrices down to the joint and link level. The stiffness distributions of the tool head over a prescribed task workspace are predicted and the contributions of joint/link compliances are evaluated using a set of global indices.

Introduction

In recent years, it has witnessed high demands for large parts in several growing industrial sectors such as aeronautics, astronautics, railroad, and shipping, etc. [1]. The classical and most frequent solution to machining very large parts is to build very large machine tools with serial travels. However, machine tools having large footprints are not suitable for the circumstances where the essential tasks are the on-site machining of relatively small features scattered in several distinct areas on large parts, hole drilling on skin of an aircraft wing, window cutting of wagon panel of a locomotive carriage, for example. A feasible and cost-effective solution is to employ a full 5-DOF (degree-of-freedom) hybrid kinematic machine [2] (or hybrid robot) which can be built as a plug-and-play robotized module mounted on a long reference track such that it can be rapidly and exactly located in the area where machining needs to be performed in situ. This statement can be exemplified by the very successful applications of the Tricept robot [3], which combines three translational parallel axes plus two serial rotary axes, allowing high rigidity and dynamic response to be achieved within a relatively large work envelope. Similar solution of large volume machine tools with hybrid architectures is the Exechon robot [4] proposed by the same inventor of the Tricept.

Stiffness is one of the most important performances of the above-mentioned hybrid robots when they are applied for high-speed machining, where high rigidity and high accuracy are crucially required. Motivated by these requirements, the last two decades have seen tremendous efforts toward this topic [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] by taking the Tricept robot as the most widely studied subject [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. For instance, Joshi and Tsai [5] compared the stiffness characteristics of the 3-DOF parallel mechanism within the Tricept robot with that of a 3-DOF manipulator having three supporting legs by merely considering the actuation compliances. Zhang and Gosselin [6], [7], [8], [9] proposed the concept of “virtual joint” for stiffness modeling by considering the bending and torsional compliances of the properly constrained passive limb. Wang et al. [12] employed the overall Jacobian to formulate the stiffness model by taking into account the compatibility conditions of the passive limb. More recently, Wang et al. [15] proposed a stiffness performance index for kineto-elastic statics analysis of parallel mechanism by which the structural optimization issue of the Tricept robot under heavy-load working conditions was investigated. As for the Exechon robot, it seems that Li et al. [16] were first to present an analytical stiffness model using screw theory and the principle of virtual work. Bi [17] formulated stiffness matrix of the Exechon X700 by which the bearings of both axial and torsional compliances of the actuated limbs on compliance of the end-effector were revealed. By exploring the substructure synthesis technique, Zhang et al. [18] developed a stiffness model by taking into account the compliances of both actuated and passive joints as well as limb structures, leading to a parametric analysis that provides useful information for the structural optimization and rigidity improvement. Wang et al. [19] developed a stiffness model by taking into account gravitational effects of the movable components of a 5-DOF hybrid robot similar to the Exechon robot. Stiffness modeling, analysis and optimization of various other parallel kinematic machines designed for large part machining were also studied in [20], [21], [22], [23], [24].

Driven by the motivation to develop new robotic structures competitive to the well-established Tricept and Exechon, a novel 5-DOF hybrid robot named TriMule was proposed in [25], [26] (see Fig. 1), which is composed of a 3-DOF R(2-RPS&RP)&UPS parallel mechanism plus a A/C wrist. Here, R, P, U, and S represent revolute, prismatic, universal, and spherical joints, respectively; and the underlined P denotes an actuated prismatic joint. The parallel mechanism comprises a spatial limb plus a 2-RPS&RP planar linkage, connected by a pair of R joints to the machine frame at either side of the base link which is elaborately designed into a three-in-one part. This feature brings a special issue that the compliance compatibility conditions amongst three planar limbs must be taken into account in the semi-analytic stiffness modeling of the parallel mechanism. The reminder of this paper is organized as follows. Having addressed the significance of stiffness modeling and given brief introduction to the TriMule robot, Section 2 presents the semi-analytic stiffness modeling strategy and procedure of the TriMule robot with particular interests in dealing with the compliance compatibility arising from the three-in-one part design, resulting in the Cartesian stiffness matrices of both the 3-DOF parallel mechanism and the A/C wrist explicitly expressed in terms of the compliance matrices down to the joint and link level. In Section 3, an example is given to illustrate the effectiveness of the proposed approach with an insight into the contributions of joint/link compliances to the rigidity of the tool head over a prescribed task workspace before conclusions are drawn in Section 4.

Section snippets

Stiffness modeling

In this section we will present a semi-analytic approach for stiffness modeling of the TriMule robot by taking into account compliances of all components. Each component is either a link or 1-DOF revolute or prismatic joint. Moreover, since the stiffness modeling issue is merely considered here, we assume that compliances of a joint are linearly elastic in nature although they would exhibit nonlinear and asymmetrical behaviors under high dynamic loads.

In order to facilitate the semi-analytic

Example

In this section, rigidity of the TriMule 600 robot over a prescribed task workspace is evaluated using the stiffness model developed in Section 2. In order to make full use of the reachable workspace, the task workspace Wt of point P is defined as the combination of a cylindrical portion and a spherical portion, as shown in Fig. 6. The dimensional parameters of the robot and the prescribed task workspace are shown in Table 1. Evaluated in the corresponding body-fixed frames, Tables A.1 and A.2

Conclusions

Mainly drawing on screw theory, this paper investigates semi-analytic stiffness modeling of the TriMule robot. The conclusions are drawn as follows:

  • (1)

    In order to facilitate the stiffness modeling of R(2-RPS&RP)&UPS parallel mechanism, the 2-RPS&RP planar linkage can be treated as a 2-DOF actuated compound joint, allowing the compliance compatibility conditions to be found first. This special treatment enables the stiffness modeling to be implemented using the standard procedure proposed by [27],

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (grants 51622508 and 51420105007) and EU H2020-RISE-ECSASDP (grant 734272).

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