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Research Papers

Designing Folding Rings Using Polynomial Continuation

[+] Author and Article Information
Andrew D. Viquerat

Research Associate
Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK

Simon D. Guest

Reader in Structural Mechanics
Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: sdg@eng.cam.ac.uk

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received December 30, 2012; final manuscript received August 8, 2013; published online December 27, 2013. Assoc. Editor: J. M. Selig.

J. Mechanisms Robotics 6(1), 011005 (Dec 27, 2013) (12 pages) Paper No: JMR-12-1218; doi: 10.1115/1.4025857 History: Received December 30, 2012; Revised August 08, 2013

Two types of foldable rings are designed using polynomial continuation. The first type of ring, when deployed, forms regular polygons with an even number of sides and is designed by specifying a sequence of orientations which each bar must attain at various stages throughout deployment. A design criterion is that these foldable rings must fold with all bars parallel in the stowed position. At first, all three Euler angles are used to specify bar orientations, but elimination is also used to reduce the number of specified Euler angles to two, allowing greater freedom in the design process. The second type of ring, when deployed, forms doubly plane-symmetric (irregular) polygons. The doubly symmetric rings are designed using polynomial continuation, but in this example a series of bar end locations (in the stowed position) is used as the design criterion with focus restricted to those rings possessing eight bars.

Copyright © 2014 by ASME
Topics: Hinges , Design , Polynomials
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Figures

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Fig. 1

Top: the deployment of a 10-bar single-hinge ring, the multiple degrees of freedom are not apparent here as the deployment has been carefully controlled, and the linkage supported by the table underneath. Bottom: the deployment of the two-hinge design. (From Ref. [27])

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Fig. 2

Top view of regular-polygonal foldable ring (in mid-deployment) with n = 10. Global coordinates are indicated.

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Fig. 3

Isometric view of individual bar in regular-polygonal ring with global coordinates indicated. Two of the planes of reflection are shown. The bar hinge vectors must remain in these planes at all times.

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Fig. 4

Regular-polygonal ring bar shown in local coordinates with hinge definitions. Note that the bar has been rotated to show the angle definitions more clearly.

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Fig. 5

Example of a ring design using the original compatibility equations (n = 10), and the angle targets given in Eq. (3).

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Fig. 6

Example of a fully feasible ring moving from deployed to stowed positions (clockwise from bottom left). It is based on the targets of Eq. (3), and its angular progression is shown in Figure 5.

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Fig. 7

Example of system designed with four {φ,θ} specifications (n = 10), and the angle targets given in Equation 5.

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Fig. 10

Second example of system designed with three {φ,θ} specifications and one {θ, ψ} (n = 10), and the angle targets given in Eq. (9).

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Fig. 11

The deployment sequence of design #1 in Figure 10 (in black, based on target set in Eq. (9)) is, coincidentally, quite similar to that in Figure 6 (in white). By focusing on the deployment of a single bar it is possible to observe the differences between the two designs. The two λl and λr angles are quite different, while the deployed states of each design appear quite similar. This is because the roll angles (φ) applied to each design bring them into closer alignment. (Note that the five poses shown here do not correspond to the target poses used in the design process)

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Fig. 9

Example of system designed with three {φ,θ} specifications and one {θ, ψ} (n = 10), and the angle targets given in Eq. (8).

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Fig. 8

Second example of system designed with four {φ,θ} specifications (n = 10), and the angle targets given in Eq. (6).

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Fig. 12

The doubly symmetric 8-bar foldable loop in the deployed configuration. Hinge inclinations to the vertical are shown. The two perpendicular planes of symmetry are shown bounding the first XY quadrant. Bar lengths l1 and l2 are also labeled.

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Fig. 13

The doubly symmetric 8-bar foldable loop in the deployed configuration. Hinge XY plane locations are shown.

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Fig. 14

Key dimensions of 8-bar in stowed configuration (a) 8-bar linkage in stowed configuration (side view). (b) 8-bar linkage in stowed configuration (top view).

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Fig. 15

Internal angles for doubly symmetric 8-bar foldable ring with arbitrarily positioned vertices. All simulations were started at the ring stowed positions. Only design # 4 was found to progress satisfactorily from the stowed to deployed configuration. (a) The four distinct solutions. (b) Detail of only feasible solution.

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Fig. 16

Folding of 8-bar with arbitrarily positioned vertices, example design # 4; clockwise from bottom left.

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Fig. 17

Internal angles for rectangular 8-bar foldable ring. All simulations were started at the ring deployed positions. Only design # 1 was found to progress satisfactorily from the deployed to stowed configuration. (a) The four distinct solutions. (b) Detail of only feasible solution.

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Fig. 18

Folding of rectangular 8-bar, example design # 1; clockwise from bottom left.

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