Creating Foldable Polyhedral Nets via Evolution Control

Yue Hao, Yun-hyeong Kim, Zhonghua Xi and Jyh-Ming Lien

Recent innovations enable robots to be manufactured using low-cost planar active material and self-folded into 3D structures. Algorithmically creating a 2D crease pattern to realize the desired 3D structure involves two steps: (1) generating unfoldings (i.e., polyhedral net), and (2) creating collision-free folding motions that take polyhedral net back to the 3D shape. It is the current practice that net design and folding motion are decoupled and treated as two independent steps. This creates a major challenge in creating self-folding machines because, given a polyhedron P, the foldability of Ps nets can vary significantly. Certain nets may not be foldable even using the most sophisticated folding motion planners. This paper presents a novel learning strategy to generate foldable nets with optimized foldability. Direct evaluation on the foldability of a net is nontrivial and can be computationally expensive. Our theoretical contribution shows that it is possible to approximate foldability combining the geometric and topological properties of a net. The experimental results show that our new unfolder will not only generate valid unfoldings but also ones that are easy to fold. Consequently, our approach makes folding much simpler in designing self-folding machines.


Synthesis of Fast and Collision-free Folding of Polyhedral Nets, Yue Hao and Yun-Hyeong Kim and Jyh-Ming Lien, Proceedings of the ACM Symposium on Computational Fabrication (SCF), Jun 2018
Web Site / Paper(pdf) / BibTeX
Creating Foldable Polyhedral Nets Using Evolution Control, Yue Hao and Yun-Hyeong Kim and Zhonghua Xi and Jyh-Ming Lien, Proceedings of the Robotics: Science and Systems Conference (RSS), Jun 2018
Web Site / Paper(pdf) / Slides(pdf) / Poster(pdf) / BibTeX


Linearly Foldable Nets

The nets shown below are easily foldable by linearly interpolating the initial and target configurations. We call such a net: linearly foldable.

Linearly foldable nets created by the proposed method.

Examples of Folding Motions


We maintain a repository to archive the unfoldings (nets) of polyhedra created by our software tools. Models will continue to be added. Alternatively, you can download the assorted linearly/uniformly foldable nets (in svg format) through the links below.


Learning Strategy

Inspired by the ideas of fitness approximation and evolution control [18], we propose a novel learning strategy to overcome two major obstacles in both net design and robot design: a large number of design parameters and expensive evaluation function. We will show that foldable net can be generated without sophisticated hypothesis models for the fitness function. Our approach is computationally efficient and requires significantly fewer evaluations in each iteration.


Overview of the proposed learning strategy. In each iteration, multiple genetic-based unfolders running in parallel (shown in the boxes) generate polyhedral nets using the assigned fitness functions {f}. The best individuals are then evaluated by the motion planner f*. An external CMA-ES optimizer then updates the fitness functions {f} according to their performance from f*. This process is repeated until the evaluated folding time converges.

Transfer Learning

Conducting training on polyhedra with 100 faces or more remains time-consuming. We employ transfer learning [34] to reduce the complexity of the problem by remeshing the polyhedron with fewer faces. The simplified mesh, though may lose details, retain most geometric characteristics. Therefore, the fitness function trained from the simplified model is used as the initial fitness function to train the original model.


An optimal fitness for Bunny-128 is obtained from training on Bunny-64. If we unfold Bunny-128 via f 64 we can obtain net (b) which shares common topological and geometrical features with the net (a) and has foldability already far better than arbitrary nets. The fitness f64 is subsequently used as initial fitness for the learning of f128.

Optimization Process Visualizer


This work was supported in part by NSF EFRI-1240459 and CNS-1205260, AFOSR FA9550-17-1-0075 and Nvidia GPU Grant Program.

We thank Eun Sook Kim, a Master teacher at Seoul Hancheon Elementary School, for including our papercraft as part of the teaching materials in her classes and helping our experiments. Some early ideas in this papers were inspired from the discussions with In-Suk Choi of Seoul National University, Takahashi Shigeo of University of Aizu, Japan, Hsu-Chun Yen of National Taiwan University.

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