Inverse design of flat-foldable thick origami with smooth curved surface

TL;DR

The paper tackles the challenge of achieving smooth curved surfaces with flat-foldable thick origami by introducing an inverse design framework that builds thick origami cells with curved exterior surfaces for generalized cylinders and cones. It models the layout as planar graphs, imposes flat-foldable constraints to prevent interference, and extrudes the design to 3D along a chosen axis, with a unbonded edge enabling folding. A key finding is that the packaging ratio $η = A/a$ increases with the number of cells, and in the continuum limit the upper bound scales linearly with cell count, with equal-area cells achieving the maximum. The approach is validated through a flight-tested deployable origami wing that demonstrates practical viability for aerospace and communication applications, highlighting potential for larger, smooth-curved deployable structures while noting current limitations to straight creases and developable surfaces.

Abstract

Origami as a deployable structure offers the unique advantage of achieving compact stowage via flat-folding while forming a well-defined surface composed of rigid panels upon deployment. However, since origami consists of flat facets, it is inherently limited in forming smooth curved surfaces upon deployment. This limitation restricts its applicability in systems where smooth curved geometries are essential for performance, such as aerospace systems and electromagnetic communication devices. Herein, we propose an inverse design methodology for thick origami that is capable of flat-folding while forming a smooth curved outer surface upon deployment. By establishing flat-foldable constraints that specify the positions of the creases, our method constructs thick origami capable of flat-folding even with panels that include a curved facet. Furthermore, by representing the origami layout as a graph, we enumerate all possible origami configurations and enable flexible design of the internal structure. Analytical results show that the optimized packaging ratio increases as the number of cells in a layout increases, indicating that the proposed design methodology provides controllability over the packaging ratio through the number of cells. Using our proposed design methodology, we fabricated a deployable origami wing and demonstrated its functionality through successful flight testing, in which the wing was subjected to aerodynamic loads. Our work proposes a new strategy for packaging smooth curved surfaces, addressing the packaging challenges encountered in aerospace and electromagnetic communications and thereby providing greater design freedom.

Figures (5)

• Figure 1: Flat-foldable thick origami structure for generalized cylinders.a Overall process of converting a generalized cylinder into a flat-foldable thick origami. The design process is performed on a plane perpendicular to the vector $\mathbf{e}$, shown as the gray plane ($xy$-plane). b Construction process of a flat-foldable thick origami cell. c Geometry of a flat-foldable thick origami cell. Blue points ($\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3, \mathbf{q}_4$) denote vertices of the embedded graph, and magenta points ($\mathbf{q}_2', \mathbf{q}_4', \mathbf{v}_1, \mathbf{v}_3$) together with dashed lines indicate creases. All vectors lie on the $xy$-plane. d Construction of the thick origami structure and its flat-folding sequence. Blue dashed lines indicate bonding interfaces, which correspond to internal edges, and the red solid line denotes the $\mathbf{v}_{1}$ crease of a representative cell. Magenta regions indicate internal facets that become coplanar in the flat-folded configuration.
• Figure 2: Flat-foldable thick origami structure for generalized cones.a Overall process of converting a generalized cone into a flat-foldable thick origami. The design process is performed on a unit sphere centered at the point $\mathbf{o}$, shown as the gray plane. b Geometry of a flat-foldable thick origami cell. Blue points ($\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3, \mathbf{p}_4$) denote vertices of the embedded graph, while teal points ($\mathbf{p}_2', \mathbf{p}_4', \mathbf{u}_1, \mathbf{u}_3$) and dashed lines indicate creases. c Construction of the thick origami structure and its flat-folding sequence. Blue dashed lines indicate bonding interfaces, which correspond to internal edges, and the red solid line denotes the $\mathbf{u}_{1}$ crease of a representative cell.
• Figure 3: Flat-foldable thick origami prototypes for various generalized cylinders and generalized cones.a Generalized cylinder with a three-leaf-clover-shaped cross-section. b Generalized cylinder with a heart-shaped cross-section. c Generalized cone with a circular cross-section (cone). d Generalized cone with an elliptical cross-section (elliptic cone).
• Figure 4: Relationship between optimal packaging ratio and the number of cells.a Conversion of a generalized cylinder with an elliptical cross-section (eccentricity 0.8) into flat-foldable thick origami structures based on graphs with 4 and 6 internal faces (4-cell and 6-cell cases). The positions of graph vertices are numerically optimized during graph embedding to achieve higher packaging ratios, yielding values of 4.2 and 6.4, respectively. b Geometry of the $i$-th thick origami cell in cross-section. The cross-sectional outline of the generalized cylinder (target shape) is highlighted in red. c (top) Plot of the optimized packaging ratio versus the number of cells $n$ for a generalized cylinder with an elliptical cross-section. Results are shown for elliptical aspect ratios of 1.0, 0.6, 0.4, and 0.2. The solid lines represent the analytical continuum-limit expression of the optimal packaging ratio, while the markers indicate local maxima obtained through numerical optimization. (bottom) Plot of the coefficient of variation of the origami cell areas $\{a_{1}, a_{2}, \ldots, a_{n}\}$ obtained from each numerical optimization. d Heatmap of the analytically obtained per-cell packaging efficiency for superellipses defined as $x=\operatorname{sgn}(\cos\theta)|\cos\theta|^{2/m}$ and $y=\gamma\,\operatorname{sgn}(\sin\theta)|\sin\theta|^{2/m}$.
• Figure 5: Design and flight test of a deployable origami wing.a Process of converting an aircraft wing with a NACA 2412 airfoil into a flat-foldable thick origami. The origami layout is deliberately selected and embedded to incorporate two vertical spars within the structure. b Deployment sequence of the origami wing. The first three stages show the spanwise folding process in reverse, while the final two stages represent the reverse of flat-folding. c Prototype UAV equipped with the deployable origami wing. The left wing is stowed, while the right wing is fully deployed for flight testing. d Flight trajectory of the UAV equipped with the deployable origami wing during flight testing.