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Machine Theory
Mechanism and Machine Theory 43 (2008) 627–640
ate/mechmt
A family of deployable polygons and polyhedra
Go¨khan Kiper a, Eres So¨ylemez a,*, . O¨ zgu¨rKisßisel b
a Mechanical Engineering Department, Middle East Technical University, 06531 Ankara, Turkey
b Mathematics Department, Middle East Technical University, 06531 Ankara, Turkey
Received 27 July 2006; received in revised form 9 April 2007; accepted 21 April 2007
Available online 19 June 2007
Abstract
A new linkage type for resizing polygonal and polyhedral shapes is proposed. The single degree-of-freedom planar link-
ages considered mainly consist of links connected by revolute joints. It is shown that the group of mechanisms obtained
realize Cardanic Motion. The polyhedral linkages proposed are constructed by implementing the proposed planar linkages
on the faces and interconnecting them by links at the vertices to retain the solid angles of the polyhedral shape of interest.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Deployable structures; Deployable polygons; Polyhedral linkages
1. Introduction
Overconstrained mechanisms constituting polyhedral shapes have recently drawn special attention of
inventors and engineers. One of the first examples to such linkages was given by Fuller [1]: the jitterbug. Later
on, many jitterbug-like linkages were discovered [2]. Hoberman [3] was the one who introduced polyhedral
linkages as worldwide known toys. Very recent polyhedral linkages were proposed by Wohlhart [4–8], Agra-
wal et al. [9] and Kova`cs et al. [10,11]. These linkages are deployable structures that find applications in 3-
dimensional object scaling [9] and virus motion modeling problems [10,11].
The main aim of all these designs is to scale polyhedral shapes, . to change their size while retaining their
shape. This task can be modeled by means of a special affine linear similarity transformation: a dilation. A
dilati