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Research Papers

Synthesis, Mobility, and Multifurcation of Deployable Polyhedral Mechanisms With Radially Reciprocating Motion

[+] Author and Article Information
Guowu Wei

Centre for Robotics Research,
School of Natural and Mathematical Sciences,
King's College London,
University of London,
London WC2R 2LS, UK
e-mail: guowu.wei@kcl.ac.uk

Yao Chen

Lecturer
National Prestress Engineering Research Center,
School of Civil Engineering,
Southeast University,
Nanjing 210096, China
e-mail: chenyao_seu@hotmail.com

Jian S. Dai

Chair in Mechanisms and Robotics
International Centre for
Mechanisms and Robotics,
MoE Key Laboratory for Mechanism Theory
and Equipment Design,
Tianjin University,
Tianjin 300072, China;
Centre for Robotics Research,
School of Natural and Mathematical Sciences,
King's College London,
University of London,
London WC2R 2LS, UK
e-mail: jian.dai@kcl.ac.uk

Central points of the four rigid triangles and the 12 rigid links are taken as 16 vertices of the contact polyhedron, and the lines through the 24 connecting joints are taken as 24 edges for the contact polyhedron.

Td symmetry group is based on the regular tetrahedron and known as achiral tetrahedral symmetry. The group has a total of 24 symmetry operations: one identity, eight rotation operation, 3 twofold rotation operations, six mirror operations, and six improper rotation operations.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received November 4, 2013; final manuscript received April 28, 2014; published online June 11, 2014. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 136(9), 091003 (Jun 11, 2014) (12 pages) Paper No: MD-13-1508; doi: 10.1115/1.4027638 History: Received November 04, 2013; Revised April 28, 2014

Extending the method coined virtual-center-based (VCB) for synthesizing a group of deployable platonic mechanisms with radially reciprocating motion by implanting dual-plane-symmetric 8-bar linkages into the platonic polyhedron bases, this paper proposes for the first time a more general single-plane-symmetric 8-bar linkage and applies it together with the dual-plane-symmetric 8-bar linkage to the synthesis of a family of one-degree of freedom (DOF) highly overconstrained deployable polyhedral mechanisms (DPMs) with radially reciprocating motion. The two 8-bar linkages are compared, and geometry and kinematics of the single-plane-symmetric 8-bar linkage are investigated providing geometric constraints for synthesizing the DPMs. Based on synthesis of the regular DPMs, synthesis of semiregular and Johnson DPMs is implemented, which is illustrated by the synthesis and construction of a deployable rectangular prismatic mechanism and a truncated icosahedral (C60) mechanism. Geometric parameters and number synthesis of typical semiregular and Johnson DPMs based on the Archimedean polyhedrons, prisms and Johnson polyhedrons are presented. Further, movability of the mechanisms is evaluated using symmetry-extended rule, and mobility of the mechanisms is verified with screw-loop equation method; in addition, degree of overconstraint of the mechanisms is investigated by combining the Euler's formula for polyhedrons and the Grübler–Kutzbach formula for mobility analysis of linkages. Ultimately, singular configurations of the mechanisms are revealed and multifurcation of the DPMs is identified. The paper hence presents an intuitive and efficient approach for synthesizing PDMs that have great potential applications in the fields of architecture, manufacturing, robotics, space exploration, and molecule research.

Copyright © 2014 by ASME
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Figures

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Fig. 1

Prototypes of regular deployable polyhedral mechanisms

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Fig. 2

A single-plane-symmetric spatial 8-bar linkage

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Fig. 3

A rectangular prism and its geometric parameters

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Fig. 4

Synthesis of a deployable rectangular prismatic mechanism

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Fig. 5

A deployable rectangular prismatic mechanism

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Fig. 6

A truncated icosahedron (C60) and its geometry

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Fig. 7

A deployable truncated icosahedral (C60) mechanism

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Fig. 8

A deployable cuboctahedral mechanism

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Fig. 9

A deployable rhombicuboctahedral mechanism

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Fig. 10

A deployable triangular prismatic mechanism

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Fig. 11

A deployable square pyramid mechanism

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Fig. 12

A deployable tetrahedral mechanism and its coordinates

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Fig. 15

Multifurcation of the tetrahedral mechanism: Case II

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Fig. 13

Singularity and its associated bifurcation

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Fig. 14

Multifurcation of the tetrahedral mechanism: Case I

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