Conical Kresling Origami: Mechanics & Design

Updated 23 August 2025

Conical Kresling Origami is a class of reconfigurable structures with conical crease patterns that enable programmable multistability and tunable mechanical behavior.
It exploits the interplay of panel bending and crease mechanics to achieve snap-through transitions, bistability, and energy absorption for advanced applications.
Numerical simulations and PINN-based inverse design methodologies facilitate precise control over mechanical responses in complex, deployable architectures.

Conical Kresling Origami (CKO) refers to a class of mechanically reconfigurable structures characterized by triangulated or polygonal crease patterns arranged on the frustum of a cone. CKO is a generalization of the canonical Kresling pattern, which is based on cylindrical symmetry, to conical geometries with either constant or graded apex angles. These structures exploit the interplay of panel bending, crease mechanics, and global geometry to realize programmable multistability, bistability, snap-through transitions, and highly tunable mechanical responses. CKO appears both as a direct subject of paper and as a central design motif in deployable metamaterials, robotic actuators, energy absorbers, adaptive electronic membranes, and personalized devices.

1. Geometric Foundations and Classification

Conical Kresling Origami structures are assemblies of triangular (or, in some cases, quadrilateral) panels connected by creases, with the panels’ vertices lying on one or more cones of revolution. There are two archetypal variants:

Classical or Cylindrical Kresling Origami: Panels are arranged between two parallel, congruent regular polygons; the structure is strictly cylindrical.
Conical Kresling Origami (CKO): The two end polygons are of different sizes and/or not parallel; the triangulation adapts to the conical frustum. The key governing parameter is the apex (“half”) angle $\lambda$ of the generating cone, with $q = \cot\lambda$ (see (Nawratil, 2021)). By varying $\lambda$ , one programs the tightness of the spiral, the longitudinal scaling, and the rate at which cross-sections change with height.

The panel network can be constructed via stacking anti-frusta, direct triangulation on a conical surface, or by group orbit procedures that enforce conical symmetry (see (Liu et al., 2023)). Edge-length assignment and closure equations guarantee that adjacent facets fit precisely and that the structure is globally compatible and free of self-intersections over admissible intervals in the design space (Nawratil, 2021).

2. Mechanics: Bistability, Nonlinearity, and Snapping

CKO exhibits profound nonlinear behavior rooted in the competition between crease (hinge) mechanics and panel bending or stretching. The minimal analytical framework emerges from the paper of isometric foldable cones (“f-cones”), with governing equations that generalize to CKOs:

$\kappa'' + (1+c)\kappa + \frac{1}{2}\kappa^3 = 0$

Here, $\kappa(s)$ is the dimensionless curvature along the crease, $c$ is a parameter related to the circumferential stress, and the cubic nonlinearity encodes the departure from linear response at large deflections (Andrade et al., 2019). The crease mechanics are modeled by an energy law,

$g(\psi) = -Lk\cos(\psi - \psi_0)$

with torsional stiffness $k$ , panel-bending modulus $B$ , and a preferred folding angle $\psi_0$ . The interplay of these terms produces a double-well energy landscape with two metastable states (“rest” and “snapped”), physically manifesting as structural bistability and discrete snap-through transitions (see (Andrade et al., 2019, Nawratil, 2021)).

In the conical context, the apex angle directly determines the snappability and the energy barriers separating the stable configurations. The “snappability index”,

$\varsigma = \frac{U_{\text{total}}}{E \cdot \text{Vol}_{\text{total}}}$

where $U_{\text{total}}$ is the total elastic energy at the “shaky” (transition) configuration, quantitatively measures how much energy density is required for a snap transition (Nawratil, 2021).

3. Numerical Modeling: Inextensibility, Phase-Field Regularization, and Isostatic Simulation

Traditional origami analysis assumes inextensible panels and infinitely thin creases, but mechanical reality—especially for CKO—demands the inclusion of stretching, bending, and thick panel effects. Two modeling strategies are prominent:

Phase-field–like models (Andrade et al., 2019): The crease is treated as a finite-width “soft” region; the model embeds a local variation (often mimicked as a temperature gradient) in material stiffness, allowing for smoothing of the crease boundary and finite bending/stretching in a narrow zone. This accommodates realistic mechanics near vertices and finite fold width.
Thick origami simulation with isostatic reduction (Micheletti et al., 28 Apr 2024): Here, a CKO is regarded as an assembly of panels linked by door hinges (pure rotation) or sliding hinges (allowing translation along the hinge axis). By judiciously introducing sliding hinges—typically at inter-module connections—global static indeterminacy is eliminated (i.e., the structure becomes isostatic, $M = 0$ , $S = 0$ ). This ensures uniqueness of internal forces and moments under loading, which is crucial for robust deployable engineering applications.

The folding kinematics are numerically realized by integrating kinematic constraints on the Euclidean group via the exponential map,

$g_{n+1} = \exp(\delta v)g_n$

and solving statically for the internal reactions in a manner consistent with imposed boundary conditions and external loads (Micheletti et al., 28 Apr 2024).

4. Programmable Multistability and PINN-Based Inverse Design

The highly nonlinear mechanical response of CKO gives rise to programmable multistability and enables the deliberate design of targeted energy landscapes. This is achieved by:

Physics-Informed Neural Network (PINN) modeling (Kang et al., 19 Aug 2025): The model embeds the mechanical equilibrium ( $\partial U/\partial\phi = 0$ ) as a hard constraint during network training. For a CKO with $n$ units, the strain energy is modeled as

$U(h, \phi) = \sum_{i=1}^n \left[ k_m (L_m(h,\phi) - L_{m,0})^2 + k_y(L_y(h, \phi) - L_{y,0})^2 \right]$

where $h$ is the vertical height, $\phi$ is the rotation, and $L_{m,y}$ are the lengths of the mountain/valley creases. For inverse design, the optimizer tunes geometric parameters ( $a$ , $b$ , $c$ , $\beta$ , $n$ ) to achieve target stable states and energy barriers.

Hierarchical CKO assemblies: The technique extends to stacked (multi-layer) CKOs, each layer individually programmed for a distinct energy barrier. This enforces sequential deployment: under increasing compression, each layer snaps in turn, triggered by its unique barrier magnitude, ensuring passively controlled, ordered actuation sequences (Kang et al., 19 Aug 2025).

PINN-based approaches are validated by finite element simulations and physical prototypes (e.g., PET-based systems), which confirm agreement between predicted and observed deployment sequences and energy landscapes.

5. Material Realization, Fabrication, and Application Spaces

CKO’s realization in engineering systems draws from advanced fabrication and judicious material choices:

Fabrication with 3D Printing and Laminated Composites: CKO implementations employ rigid/flexible hybrids—e.g., a rigid photopolymer core with a compliant, rubber-like frame at creases (Khazaaleh et al., 2021), or thick origami assemblies manufactured with heat-sealed fabrics (Liu et al., 30 Jan 2025). Polyjet and FDM 3D printing permit precise control of crease (hinge) compliance, panel thickness, and geometric gradients.
Mechanical Tuning by Geometric and Material Parameters: The mechanical response—including stiffness, bistability, hardening/softening profile, and multistability—is programmable by (i) modulating apex angle $\lambda$ , (ii) adjusting mountain/valley crease lengths, facet size, and number of units $n$ , and (iii) inserting small vertex cuts to tailor stiffness and the force limit point (Yang et al., 2022).
Group Orbit Lagrangian Design: Systematic design leverages Lagrangian and group orbit methods—solving Frenet–Serret ODEs for curved creases and applying circle or conformal group actions to tile unit cells around a conical vertex—to ensure geometric compatibility and energy scaling ( $E \sim h^3$ ) (Liu et al., 2023).
Sensing, Robotics, Energy Absorption, and Deployability: CKO serves as the enabling mechanism in deployable reflectors, robotic manipulators (where bi-stability or multi-stability allows discrete “joints”), energy absorbers (exploiting translation–rotation coupling for compact shock mitigation), adaptive electronic membranes (where the coupled twist–extension is used to switch between sensor modalities), and customized orthoses (tailoring conical segments for personalized fit and compliance) (Yao et al., 11 Jun 2024, Khazaaleh et al., 2023, Liu et al., 30 Jan 2025).

6. Dynamic, Topological, and Chaotic Behaviors

CKO exhibits rich, physics-governed dynamic phenomena:

Deployment Dynamics and Snap-Through: Full six-degree-of-freedom models reveal sensitivity to initial conditions, geometric parameters, and damping. Both axial and off-axis dynamic responses are critical: the axial-twist mode determines deployment, while off-axis modes relate to robustness and disturbance rejection (Kidambi et al., 2020).
Topological State Transfer: Lattices constructed from CKO-like Kresling dimers (alternating chirality) support twist-induced modulations in their band structure, resulting in topologically nontrivial state transfer between edges. The Zak phase quantifies topology, with quasi-static twist effecting phase transitions and enabling in-situ energy routing or robust boundary localization (Miyazawa et al., 2022),

$\phi_m = -\sum_k \text{Im} \ln \left< u_k^{(m)} | u_{k+1}^{(m)} \right>$

Geometry-Informed Dynamic Mode Decomposition: For giant parameter spaces or intricate coupling (as in dual Kresling or CKO), augmented data-driven models (giDMD) employ regressor architectures enriched by geometric information ( $h$ , $a$ , $b$ , $\Psi$ , orientation angles) to improve predictive accuracy for periodic and low-frequency dynamics, though chaotic regimes remain challenging (Li et al., 2023).

7. Limitations and Current Research Frontiers

CKO, while powerful, is subject to several design and modeling constraints:

The analytic treatment relies on idealized assumptions (e.g., panel inextensibility, perfect crease geometry) whose relaxation reveals strong sensitivity to small perturbations (Andrade et al., 2019, Yang et al., 2022).
Fabrication defects and manufacturing process limitations (precise fold width, panel thickness, etc.) can produce large deviations in global stiffness and snap-through thresholds.
Computational models rapidly increase in complexity for thick or multimodular CKO, requiring advanced numerical methods (e.g., large-deformation shell FEA, phase-field regularization).
PINN and giDMD methods, while effective for periodic and multi-stable regimes, have reduced generalizability in highly chaotic or regime-shifting dynamics. The integration of further physical constraints and global loss regularization remains an active topic (Kang et al., 19 Aug 2025, Li et al., 2023).

A plausible implication is that future work will center on combined analytical–numerical design pipelines, improved material models, and increasingly precise inverse-design routines, underpinned by fully data-free or hybrid PINN approaches.

Summary Table: Governing Quantities and Design Tools

Parameter/Method	Role in CKO Mechanics or Design	Reference(s)
Apex angle ( $\lambda$ ), $q = \cot \lambda$	Controls conical shape and snap barrier	(Nawratil, 2021, Liu et al., 2023)
Nonlinear elastica equation ( $\kappa''+(1+c)\kappa+ \tfrac{1}{2}\kappa^3=0$ )	Captures facet/crease competition, bistability	(Andrade et al., 2019)
Snappability index ( $\varsigma$ )	Quantifies energy density for snap	(Nawratil, 2021)
Phase-field regularization	Models realistic crease width, bending/stretch	(Andrade et al., 2019)
PINN loss: $L = \sum_i (U(h_i,\phi(h_i)) - U_\text{target}(h_i))^2 + \lambda_\text{EL} (\partial_\phi U - 2A_\text{reg}\partial_h^2 \phi)^2$	Combines equilibrium enforcement with target landscape	(Kang et al., 19 Aug 2025)
Group orbit procedure	Ensures global symmetry and tiling on cone	(Liu et al., 2023)

CKO thereby stands as a paradigmatic example of geometric mechanics in functional origami, unifying principles of nonlinear elasticity, symmetry, programmable multistability, and hierarchical design in mechanical metamaterials and deployable structures.