Research Papers

Designing Optimal Origami Structures by Computational Evolutionary Embryogeny

[+] Author and Article Information
Wei Li

Department of Mechanical Engineering,
Texas A&M University,
3123 TAMU,
College Station, TX 77843-3123
e-mail: liwei.meen@gmail.com

Daniel A. McAdams

Department of Mechanical Engineering,
Texas A&M University,
3123 TAMU,
College Station, TX 77843-3123
e-mail: dmcadams@tamu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received May 30, 2014; final manuscript received December 2, 2014; published online January 30, 2015. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 15(1), 011010 (Mar 01, 2015) (11 pages) Paper No: JCISE-14-1194; doi: 10.1115/1.4029561 History: Received May 30, 2014; Revised December 02, 2014; Online January 30, 2015

As the advantages of foldable or deployable structures are being discovered, research into origami engineering has attracted more focus from both artists and engineers. With the aid of modern computer techniques, some computational origami design methods have been developed. Most of these methods focus on the problem of origami crease pattern design—the problem of determining a crease pattern to realize a specified origami final shape, but do not provide computational solutions to actually developing a shape that meets some design performance criteria. This paper presents a design method that includes the computational design of the finished shape as well as the crease pattern. The origami shape will be designed to satisfy geometric, functional, and foldability requirements. This design method is named computational evolutionary embryogeny for optimal origami design (CEEFOOD), which is an extension of the genetic algorithm (GA) and an original CEE. Unlike existing origami crease pattern design methods that adopt deductive logic, CEEFOOD implements an abductive approach to progressively evolve an optimal design. This paper presents how CEEFOOD—as a member of the GA family—determines the genetic representation (genotype) of candidate solutions, the formulation of the objective function, and the design of evolutionary operators. This paper gives an origami design problem, which has requirements on the folded-state profile, position of center of mass, and number of creases. Several solutions derived by CEEFOOD are listed and compared to highlight the effectiveness of this abductive design method.

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Fig. 1

Workflow procedures for two types of origami design methods. (a) Deductive and (b) abductive.

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Fig. 2

(a) A crease pattern represented by the locations of the vertices and the creases that link in between; (b) the mutation on one bit in the genetic code for the first intuitive geometric representation of (a) results in an invalid crease pattern with a breakage; and (c) the mutation on one bit in the genetic code for the second intuitive geometric representation of (a) results in an invalid crease pattern due to the effective loss of one crease

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Fig. 3

(a) A PMR of a square origami sheet, where each circle represents one “unlit” LED cell; (b) to use a PMR to represent an origami, the faces must be dyed, so that the two faces on two sides of every crease must be different; and (c) the LED cells in PMR are assigned with light colors and light radius to make the “LED matrix” origami sheet look as (b)

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Fig. 4

An iterative cycle of CEEFOOD

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Fig. 5

The workflow of CEEFOOD

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Fig. 6

The individual development of a PMR with 3 × 3 cells. (a) The embryonic state of the PMR has five blue cells and four red cells; (b) the rule directs two cells to change their colors and guilds the PMR to an intermediate state; and (c) the rules guilds the PMR to the mature state, which also has five blue cells and four red cells.

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Fig. 7

The individual development of four equivalent instances. For having a switch-off rule, (c) and (d) are both nonaging instances.

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Fig. 8

The code assimilation expands the search region

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Fig. 9

Initial guess pattern used for the first experiment group. (a) The initial guess has five creases; (b) fill the faces of initial guess with blue and red; (c) a replica of the initial guess that will be used by CEEFOOD to initiate the evolution; (d) the design No. 1 with the creases numbers from 1 to 8; (e) manual folding model of the design; and (f) the flat-folded profile of the design No. 1 according to the folding in (e).

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Fig. 10

Folding procedure for deriving the folded shape in Fig. 9(e) from its crease pattern Fig. 9(d). (a) Print out the crease pattern; (b) fold the lower right corner inward along crease No. 5; and (c) creases other than No. 5 are linked, thus must be folded simultaneously. (d) The creases are further folded, and the right half portion of the origami sheet will be folding underneath the left half portion. When all the creases are folded to ±180 deg, the shape will become to the state shown in Fig. 9(e).

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Fig. 11

Two crease pattern designs from the second experiment group, which uses randomly generated initial generation of candidate solutions. In this figure, (a)–(c) present the crease pattern of design No. 2 and its flat-folded profile; (d)–(f) present the crease pattern of design No. 3 and its flat-folded profile.

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Fig. 12

The OFV of the topmost elite through generations



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