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Research Papers

Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Plane-Symmetric Case

[+] Author and Article Information
P. C. López-Custodio

Centre for Robotics Research,
King's College London,
University of London,
Strand, London WC2R 2 LS, UK
e-mail: pablo.lopez-custodio@kcl.ac.uk

J. S. Dai

Centre for Robotics Research,
King's College London,
University of London,
Strand, London WC2R 2 LS, UK
e-mail: jian.dai@kcl.ac.uk

J. M. Rico

Mechanical Engineering Department,
DICIS, Universidad de Guanajuato,
Salamanca 36885, Guanajuato, Mexico
e-mail: jrico@ugto.mx

1Corresponding author.

Manuscript received August 16, 2017; final manuscript received December 21, 2017; published online March 1, 2018. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 10(3), 031002 (Mar 01, 2018) (12 pages) Paper No: JMR-17-1255; doi: 10.1115/1.4039002 History: Received August 16, 2017; Revised December 21, 2017

This paper for the first time reveals a set of special plane-symmetric Bricard linkages with various branches of reconfiguration by means of intersection of two generating toroids, and presents a complete theory of the branch reconfiguration of the Bricard plane-symmetric linkages. An analysis of the intersection of these two toroids reveals the presence of coincident conical singularities, which lead to design of the plane-symmetric linkages that evolve to spherical 4R linkages. By examining the tangents to the curves of intersection at the conical singularities, it is found that the linkage can be reconfigured between the two possible branches of spherical 4R motion without disassembling it and without requiring the usual special configuration connecting the branches. The study of tangent intersections between concentric singular toroids also reveals the presence of isolated points in the intersection, which suggests that some linkages satisfying the Bricard plane-symmetry conditions are actually structures with zero finite degrees-of-freedom (DOF) but with higher instantaneous mobility. This paper is the second part of a paper published in parallel by the authors in which the method is applied to the line-symmetric case.

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Topics: Linkages
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Figures

Grahic Jump Location
Fig. 1

The general plane-symmetric case of Bricard linkages

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Fig. 2

An RR dyad generating a singular toroid (r=l,s=0)

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Fig. 3

The concentric singular toroids intersection: (a) surfaces and (b) resultant Bricard plane-symmetric linkages

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Fig. 4

Common perpendicular diagram for the plane-symmetric linkages

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Fig. 5

Two singular toroids that are tangent to each other in the XAZA plane

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Fig. 6

A Bricard plane-symmetric linkage with finite mobility zero in its two different assembly modes

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Fig. 7

A Bricard plane-symmetric linkage discovered with finite mobility zero in the assembly mode with E(q1)=P1

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Fig. 8

A case with two singular curves in the intersection: (a) BA and BB curves and (b) surfaces and intersection

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Fig. 9

A reconfigurable Bricard plane-symmetric linkage in its spherical 4R operation mode, with its two secondary circles tangent to each other and to curve C1 and the linkage is about to escape to V1

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Fig. 10

A plane-symmetric configuration of a spherical 4R linkage with two opposite links of the same angle

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Fig. 11

Tangent vectors to the intersection curves at the double singularity, for the three possible cases: rA > rB, rA = rB, and rA < rB

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Fig. 12

Two cases of branch reconfiguration flowcharts when the intersection of concentric singular toroids is composed of two singular curves: (a) rA < rB and γA≠γB (example presented in this subsection) and (b) rA = rB and |γA|=|γB|

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Fig. 13

A reconfigurable Bricard plane-symmetric linkage that allows the reconfiguration between two spherical 4R branches through two Bricard branches. The intersection of the concentric singular toroids is composed of two singular curves.

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Fig. 14

Several configurations of the plane-symmetric Bricard linkages when the intersection of concentric toroids contains two singular curves

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