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Research Papers

# Analysis of Two Spherical Parallel Manipulators With Hidden Revolute Joints

[+] Author and Article Information
Ju Li

School of Mechanical Engineering,
Changzhou University,
Changzhou, Jiangsu 213164, China
e-mail: wangju0209@163.com

J. Michael McCarthy

Department of Mechanical and
Aerospace Engineering,
University of California, Irvine,
Irvine, CA 92697
e-mail: jmmccart@uci.edu

1Corresponding author.

Manuscript received July 26, 2016; final manuscript received December 13, 2016; published online March 22, 2017. Assoc. Editor: Shaoping Bai.

J. Mechanisms Robotics 9(3), 031007 (Mar 22, 2017) (10 pages) Paper No: JMR-16-1215; doi: 10.1115/1.4035542 History: Received July 26, 2016; Revised December 13, 2016

## Abstract

In this paper, we examine two spherical parallel manipulators (SPMs) constructed with legs that include planar and spherical subchains that combine to impose constraints equivalent to hidden revolute joints. The first has supporting serial chain legs constructed from three revolute joints with parallel axes, denoted R$∥$R$∥$R, followed by two revolute joints that have intersecting axes, denoted $RR̂$. The leg has five degrees-of-freedom and is denoted R$∥$R$∥$R-$RR̂$. Three of these legs can be assembled so the spherical chains all share the same point of intersection to obtain a spherical parallel manipulator denoted as 3-R$∥$R$∥$R-$RR̂$. The second spherical parallel manipulator has legs constructed from three revolute joints that share one point of intersection, denoted $RRR̂$, and a second pair of revolute joints with axes that intersect in a different point. This five-degree-of-freedom leg is denoted $RRR̂$-$RR̂$. The spherical parallel manipulator constructed from these legs is 3-$RRR̂$-$RR̂$. We show that the internal constraints of these two types of legs combine to create hidden revolute joints that can be used to analyze the kinematics and singularities of these spherical parallel manipulators. A quaternion formulation provides equations for the quartic singularity varieties some of which decompose into pairs of quadric surfaces which we use to classify these spherical parallel manipulators.

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Topics: Manipulators , Chain

## Figures

Fig. 1

The spherical parallel manipulator, 3-R∥R∥R-RR̂, consisting of three revolute joints that form a planar chain followed by two revolute joints with axes that intersect. The three legs are assembled so the axes of the end pairs of revolute joints intersect at O′.

Fig. 2

The spherical parallel manipulator, 3-RRR̂-RR̂, formed from three revolute joints with axes that intersect in one point and two more revolute joints with axes that intersect. The three legs are assembled so the axes of the end pairs of revolute joints intersect at O′.

Fig. 3

Spherical parallel manipulators by Kong and Gosselin [9] that have the same structure as the 3-R∥R∥R-RR̂ and 3-RRR̂-RR̂, denoted (a) 3-(RRR)E-ŘŘ and (b) 3-(RRR)S-ŘŘ

Fig. 4

The link R3R4 connects the planar and spherically constrained portions of the 3-R∥R∥R-RR̂

Fig. 5

The platform is supported by three spherical 3R chains RiHRi4Ri5, and each base link Ri4RiHRi3 is driven by the planar four-bar linkage Ri1Ri2Ri3RiH. The revolute joints RiH, i = 1,2,3, denote the hidden revolute joints.

Fig. 6

The platform is supported by three spherical 3R chains RiHRi4Ri5, and each base link Ri4RiHRi3 is driven by the spherical 4R linkage Ri1Ri2Ri3RiH. The revolute joints RiH, i = 1,2,3, denote the hidden revolute joints.

Fig. 7

The 3-RRR̂ manipulator. ui, wi, vi, i = 1,2,3, denote the unit vectors along the axes of joints RiH, Ri4, and Ri5, respectively, and μi is the angle between axes ui and wi, while τi is the angle between axes wi and vi.

Fig. 8

Type 1: (a) the pivots ui lie on a great circle, (b) the pivots vi lie on a great circle, and (c) both sets of pivots ui and vi lie on great circles

Fig. 9

Type 2: (a) two of the pivots vi on the moving body are coincident, (b) two of the pivots ui on the fixed body are coincident, and (c) there are two sets of coincident pivots on both the moving and fixed bodies

Fig. 10

Type 3: (a) the pivots vi are on a great circle and two fixed pivots are coincident and (b) the pivots ui are on a great circle and two of the moving pivots are coincident

Fig. 11

The singularity variety of the 3-RRR̂ spherical parallel manipulator that has both sets of pivots ui and vi lie on great circles

Fig. 12

The singularity variety of the 3-RRR̂ spherical parallel manipulator that has two coincident pivots on the base and two coincident pivots on the platform, decomposes into pairs of quadric surfaces

Fig. 13

The singularity variety of the 3-RRR̂ spherical parallel manipulator whose pivots ui are on a great circle and two of the moving pivots are coincident, decomposes into pairs of quadric surfaces

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