Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages

Chao-Yang Song; Yan Chen; I-Ming Chen

doi:10.1115/1.4026339

Journal of Mechanisms and Robotics | Volume 6 | Issue 3 | research-article

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Research Papers

Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages

Chao-Yang Song, Yan Chen ¹ and I-Ming Chen

[+] Author and Article Information

Chao-Yang Song

Engineering Product Development Pillar,
Singapore University of Technology and Design,
20 Dover Drive,
Singapore 138682, Singapore
e-mail: chaoyang_song@sutd.edu.sg

Yan Chen

Professor
Key Laboratory of Mechanism
Theory and Equipment Design of
Ministry of Education,
School of Mechanical Engineering,
Tianjin University,
Tianjin 300072, China
e-mail: yan_chen@tju.edu.cn

I-Ming Chen

Associate Professor
School of Mechanical and
Aerospace Engineering,
Nanyang Technological University,
50 Nanyang Avenue,
Singapore 639798, Singapore
e-mail: michen@ntu.edu.sg

¹Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 21, 2013; final manuscript received October 17, 2013; published online April 3, 2014. Assoc. Editor: Philippe Wenger.

J. Mechanisms Robotics 6(3), 031002 (Apr 03, 2014) (10 pages) Paper No: JMR-13-1057; doi: 10.1115/1.4026339 History: Received March 21, 2013; Revised October 17, 2013

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Abstract

Abstract | Introduction | The Solutions of Closure Equations of the Original General Line-Symmetric Bricard Linkage | The Solutions of Closure Equations of the Revised General Line-Symmetric Bricard Linkage | Relationship Between the Original and Revised General Line-Symmetric Bricard Linkages | Conclusion and Discussions | Acknowledgements | References

In this paper, the solutions to closure equations of the original general line-symmetric Bricard 6R linkage are derived through matrix method. Two independent linkage closures are found in the original general line-symmetric Bricard 6R linkage, which are line-symmetric in geometry conditions, kinematic variables and spatial configurations. The revised general line-symmetric Bricard 6R linkage differs from the original linkage with negatively equaled offsets on the opposite joints. Further analysis shows that the revised linkage is equivalent to the original linkage with different setups on joint axis directions. As a special case of the general line-symmetric Bricard linkage, the line-symmetric octahedral Bricard linkage also has two forms in the closure equations. Their closure curves are not independent but joined into a full circle. This work offers an in-depth understanding about the kinematics of the general line-symmetric Bricard linkages.

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References

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Bricard, R., 1897, “Mémoire sur la théorie de l'octaèdre articulé,” J. Pure Appl. Math., 3, pp. 113–150 (English translation by E. A. Coutsias, 2010, e-print, http://arxiv.org/abs/1203.1286).

Figures

Fig. 1

The setup of the Denavit and Hartenberg's parameters

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Fig. 2

The kinematic paths of the original Form I general line-symmetric Bricard 6R linkage

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Fig. 3

The kinematic paths of the original Form II general line-symmetric Bricard 6R linkage

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Fig. 4

The spatial configuration of the original Form I general line-symmetric Bricard 6R linkage when θ1I=π/3

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Fig. 5

The spatial configuration of the original Form II general line-symmetric Bricard 6R linkage when θ1II=π/3

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Fig. 6

The SVD results of the original general line-symmetric Bricard 6R linkages: (a) Form I linkage and (b) Form II linkage

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Fig. 7

The kinematic paths of the revised Form I' general line-symmetric Bricard linkage

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Fig. 8

The kinematic paths of the revised Form II' general line-symmetric Bricard 6R linkage

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Fig. 9

The spatial configuration of the revised Form I′ general line-symmetric Bricard 6R linkage when θ1I'=π/3

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Fig. 10

The spatial configuration of the revised Form II' general line-symmetric Bricard 6R linkage when θ1II'=π/3

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Fig. 11

The illustrations of the general line-symmetric Bricard linkage: (a) the original Form I linkage and (b) the revised Form I′ linkage

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Fig. 12

The kinematic paths of the general line-symmetric octahedral Bricard linkage. Here, the geometric parameters are the same as Eq. (27) with ai(i+1)=0. The black solid line is from the closure equations of the Form I linkage; the grey solid line is from those of the Form II linkage.

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Song C, Chen Y, Chen I. Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages. ASME. J. Mechanisms Robotics. 2014;6(3):031002-031002-10. doi:10.1115/1.4026339.

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Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages

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