Research Papers

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

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

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

J. M. Rico

Mechanical Engineering Department,
DICIS, Universidad de Guanajuato,
Salamanca 36500, Guanajuato, Mexico
e-mail: jrico@ugto.mx

1Corresponding author.

Manuscript received June 22, 2017; final manuscript received December 14, 2017; published online March 23, 2018. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 10(3), 031003 (Mar 23, 2018) (11 pages) Paper No: JMR-17-1188; doi: 10.1115/1.4038981 History: Received June 22, 2017; Revised December 14, 2017

This paper for the first time investigates a family of line-symmetric Bricard linkages by means of two generated toroids and reveals their intersection that leads to a set of special Bricard linkages with various branches of reconfiguration. The discovery is made in the concentric toroid–toroid intersection. By manipulating the construction parameters of the toroids, all possible bifurcation points are explored. This leads to the common bi-tangent planes that present singularities in the intersection set. The study reveals the presence of Villarceau and secondary circles in the toroid–toroid intersection. Therefore, a way to reconfigure the Bricard linkage to a pair of different types of Bennett linkage is uncovered. Further, a linkage with two Bricard and two Bennett motion branches is explored. In addition, the paper reveals the Altmann linkage as a member of the family of special line-symmetric Bricard linkage studied in this paper. The method is applied to the plane-symmetric case in the following paper published together with this paper.

Copyright © 2018 by ASME
Topics: Linkages
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Grahic Jump Location
Fig. 2

An RR dyad generating a general toroid

Grahic Jump Location
Fig. 3

The line-symmetric Bricard linkages that can be analyzed as the intersection of two concentric toroids

Grahic Jump Location
Fig. 1

The general line-symmetric case of Bricard linkage

Grahic Jump Location
Fig. 4

Diagram for the line-symmetric case with sA=sB=0. (a) Assembly with vA=−vB and (b) assembly with vA = vB.

Grahic Jump Location
Fig. 9

Different configurations for an example of line-symmetric linkage where the generated toroids intersect in two circles

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Fig. 10

The four curves of uA versus uB obtained from the motion branches of the linkage. Bifurcation points are located in relation with Fig. 8. (a) Toroidal representation and (b) Cartesian plot.

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Fig. 5

Intersection of toroids A and B and their bi-tangent plane with the XZ plane

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Fig. 6

Cases with circles (in bold curves) in the intersection. (a) Villarceau circles and (b) secondary circles. In both cases, points 1–4 are permanent points of tangency. In (b) 5 and 6 are points of tangency in the XZ plane.

Grahic Jump Location
Fig. 8

An example of reconfigurable Bricard linkage, which can evolve to a pair different types of Bennett linkage. The intersection of the two concentric toroids is composed of four circles Ci, i=1,…,4. The bifurcation points are labeled as Pi, i=1,…,6. The linkage is shown in a singular configuration with E = P5.

Grahic Jump Location
Fig. 7

A toroid and a rotated copy of itself sharing secondary circles: (a) θ=2γ* and (b) θ=π−2γ*



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