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Research Papers

# Classification of Screw Systems Composed of Three Planar Pencils of Lines for Singularity Analysis of Parallel Mechanisms1

[+] Author and Article Information
Xianwen Kong

School of Engineering and Physical Sciences,
Heriot-Watt University,
Edinburgh EH14 4AS, UK
e-mail: x.kong@hw.ac.uk

Andrew Johnson

School of Engineering and Physical Sciences,
Heriot-Watt University,
Edinburgh EH14 4AS, UK

The original version of this paper was presented at the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Aug. 12–15, 2012, Chicago, IL, Paper No. DETC2012-70636.

For alternative classification of singularities, see Refs. [1,3,4].

2Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 8, 2013; final manuscript received November 16, 2013; published online March 6, 2014. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 6(2), 021008 (Mar 06, 2014) (10 pages) Paper No: JMR-13-1107; doi: 10.1115/1.4026340 History: Received June 08, 2013; Revised November 16, 2013

## Abstract

Screw systems composed of (the sum of) three planar pencils of lines are closely related to the singularity analysis of a number of three-legged parallel manipulators (PMs) in which the passive joints in each leg are a spherical joint and a single-DOF (degree of freedom) kinematic joint or generalized kinematic joint. This paper systematically classifies the screw systems composed of three planar pencils of lines based on the intersection of two planar pencils of lines, the classification of screw systems of order 2, and the reciprocal screw system of the three planar pencils of lines. The classification in this paper is more comprehensive than those in the literature. The above results are illustrated using CAD figures. This work may help readers better understand the geometric characteristics of singular configurations of a number of three-legged parallel manipulators.

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Topics: Screws
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## Figures

Fig. 1

A 2-RPS-1-SPR parallel mechanism

Fig. 2

Reciprocal screws

Fig. 3

A PPL

Fig. 4

Reciprocal screw systems of three 4-$0-systems: (a) S⊥ is a 1-$-1-$0-system, (b) S⊥ is a special 2-$0-system, and (c) S⊥ is a general 2-$0-system Fig. 5 Intersection of two PPLs: (a) Case I of a 1-$0-system, (b) Case II of a 1-$0-system, (c) Case III of 1-$0-system, and (d) Empty case

Fig. 6

Characteristics of case 4c 2-PPL-system of order 4

Fig. 7

Construction of a 3-PPL-system of order 5 using Eq. (4): (a) Case 5b, (b) Case 5c, (c) Case 5d, and (d) Case 5e

Fig. 8

Construction of a 3-PPL-system of order 5 using Eq. (3)

Fig. 9

Singularity analysis of a 2-RPS-1-SPR parallel manipulator: (a) Wrench system and plane of constraint force pencil of an RPS leg, and (b) Manipulator in a singular configuration

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