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Research Papers

Construction, Mobility Analysis and Synthesis of Polyhedra With Articulated Faces

[+] Author and Article Information
Thierry Laliberté

Département de génie mécanique,
Université Laval,
Québec QC G1V 0A6, Canada
e-mail: thierry@gmc.ulaval.ca

Clément Gosselin

Département de génie mécanique,
Université Laval,
Québec QC G1V 0A6, Canada
e-mail: gosselin@gmc.ulaval.ca

The terms sides and corners are used for polygons to distinguish them from edges and vertices, which are used for polyhedra [12].

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July 4, 2013; final manuscript received October 8, 2013; published online December 27, 2013. Assoc. Editor: Pierre M. Larochelle.

J. Mechanisms Robotics 6(1), 011007 (Dec 27, 2013) (11 pages) Paper No: JMR-13-1126; doi: 10.1115/1.4025859 History: Received July 04, 2013; Revised October 08, 2013

The concept of polyhedra with articulated faces is investigated in this paper. Polyhedra with articulated faces can be described as polyhedral frameworks, whose faces are constrained to remain planar. A mechanical arrangement based on a single type of component is proposed for the construction of the polyhedra. Then, the determination of their infinitesimal and full-cycle mobility is addressed. In some cases, they are rigid structures while in others they are articulated mechanisms. Finally, examples are given, using simulation and physical models, and several new families of articulated polyhedra are synthesized.

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References

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Figures

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Fig. 1

Mechanical component used to build PAFs: concept (left), practical implementation (right)

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Fig. 2

Mechanical assembly of a cube with articulated planar faces: concept (left), physical model (right)

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Fig. 3

Face loop of a PAF

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Fig. 4

Vertex loop of a PAF

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Fig. 5

Cube in flexed configurations: concept (left), physical model (right)

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Fig. 6

Simulation model of a cuboctahedron in the reference configuration (left) and in two flexed configurations

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Fig. 7

Physical model of a truncated octahedron in the reference configuration (left) and in a flexed configuration (right)

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Fig. 8

Physical model of a rhombicuboctahedron in the reference configuration (left) and in a flexed configuration (right)

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Fig. 9

Physical model of a great rhombicuboctahedron in the reference configuration (left) and in a flexed configuration (right)

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Fig. 10

Physical models of the truncated icosahedron (Buckyball) (left) and of the truncated dodecahedron (right)

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Fig. 11

Simulation model of a rhombicosidodecahedron in the reference configuration (left) and in a flexed configuration (right)

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Fig. 12

Simulation model of a great rhombicosidodecahedron in the reference configuration (left) and in a flexed configuration (right)

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Fig. 13

Physical models of a triangular prism in a flexed configuration (left) and of a pentagonal prism in a flexed configuration (right)

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Fig. 14

Physical model of a rhombic dodecahedron (4 zones) in two flexed configurations

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Fig. 15

Simulation models of irregular polyhedra with topologies equivalent to the ones of an octahedron (left) and of a cube (right)

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Fig. 16

Physical model of a triangular cupola in the reference configuration (left) and in a flexed configuration (right)

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Fig. 17

Simulation model of a square cupola in the reference configuration (left) and in a flexed configuration (right)

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Fig. 18

Simulation models of an elongated square pyramid (J08), an elongated square dipyramid (J15) and an elongated pentagonal gyrobirotunda (J43) in flexed configurations

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Fig. 19

Physical model of a gyrobifastigium in the reference configuration (left) and in a flexed configuration (right)

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Fig. 20

Simulation model of a triangular orthobicupola (J27) in the reference configuration (left) and in two flexed configurations

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Fig. 21

Simulation model of a pentagonal orthobicupola (J30) in the reference configuration (left) and in two flexed configurations, in which deformation and translation of the pentagonal faces can be observed

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Fig. 22

Simulation model of a pentagonal gyrobicupola (J31) in the reference configuration (left) and in two flexed configurations, in which rotations of the pentagonal faces can be observed

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Fig. 23

Simulation model of a parabiaugmented hexagonal prism (J55) in the reference configuration (left) and in two flexed configurations. Each of the flexed configurations is in a different branch of full-cycle mobility. One branch allows 1-DOF translation and 2-DOF deformations of the hexagonal face, while the other branch allows 3-DOF deformations of the hexagonal face

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Fig. 24

Simulation model of a biaugmented truncated cube (J67) in the reference configuration (left) and in two flexed configurations. Each of the flexed configurations is in a different branch of full-cycle mobility.

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Fig. 25

Simulation model of a diminished rhombicosidodecahedron (J76) in a flexed configuration. Two opposite views of the same configuration are illustrated. It is observed that a ring opposite to the diminution still lies in a plane. This explains why a second opposite diminution, as in the parabidiminished rhombicosidodecahedron (J80), does not decrease the mobility of the PAF. Also, it is interesting to observe the symmetry of the deformations.

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