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Technical Brief

# Displacement Analysis of Foldable V-Polyhedra Based on Dual Quaternions

[+] Author and Article Information
Song Lin

Tongji University,
Shanghai 201804, China
e-mail: slin@tongji.edu.cn

Jingshuai Liu

School of Mechanical Engineering,
Tongji University,
Shanghai 201804, China
e-mail: 1210295@tongji.edu.cn

Yu Zhang

School of Mechanical Engineering,
Tongji University,
Shanghai 201804, China
e-mail: sjzzy1991@163.com

Hanchao Wang

School of Mechanical Engineering,
Tongji University,
Shanghai 201804, China
e-mail: whc120005@126.com

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 21, 2017; final manuscript received May 23, 2017; published online July 10, 2017. Assoc. Editor: Ettore Pennestri.

J. Mech. Des 139(9), 094501 (Jul 10, 2017) (5 pages) Paper No: MD-17-1164; doi: 10.1115/1.4037110 History: Received February 21, 2017; Revised May 23, 2017

## Abstract

V-polyhedra is a Kokotsakis-type flat foldable rigid origami with increasing application in the engineering field. Currently, researches on origami mainly focused on foldability and mobility. In order to apply V-polyhedra in practical engineering, the analysis of kinematic characteristics is in need. This paper presents a displacement analysis methodology for the generic point belonging to any surfaces of foldable V-polyhedra. The rigid foldability of four-faced V-polyhedra and that of nine-faced V-polyhedra were discussed first. Then, the corresponding mathematical models are established with the rotating vector model constructed by dual quaternions. Finally, the correctness of the proposed method is verified through application of a symmetric pair of nine-faced V-polyhedra.

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## Figures

Fig. 1

Two cases of V-polyhedra: (a) four-faced with single vertex and (b) nine-faced with four vertexes

Fig. 2

Kinematic model of single vertex V-polyhedra: (a) four-faced V-polyhedra and (b) equivalent spherical 4-bar mechanism

Fig. 3

Rotation sequence for the displacement analysis of four-faced V-polyhedra

Fig. 4

Kinamatic model of V-polyhedra with four vertexes: (a) nine-faced V-polyhedra and (b) edge vector notations and a coupler point M on S5

Fig. 5

Rotation sequence for the displacement analysis of nine-faced V-polyhedra

Fig. 6

Solidworks model of the foldable car roof on three states [17]: (a) unfolded state, (b) intermediate state, and (c) folded state

Fig. 7

Kinematic simulation of nine-faced V-polyhedra for convertible roof in GeoGebra: (a) unfolded state φ = 135 deg, and (b) intermediate state φ = 70 deg

Fig. 8

Nodal displacement analysis results: (a) 3D trajectories and (b) projection on horizontal plane

## Errata

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