Hierarchical CKO Assemblies

• Hierarchical CKO assemblies are engineered multi-layer origami structures programmed with distinct energy barriers for controlled, sequential deployment.
• A physics-informed neural network framework predicts and designs mechanical energy landscapes by embedding equilibrium conditions and regularization into its loss function.
• Validation against finite element analysis and experiments shows high prediction accuracy (R² > 0.93) and reliable sequential actuation in complex multiscale systems.

Hierarchical CKO assemblies refer to engineered structures composed of stacked or otherwise spatially organized units of conical Kresling origami (CKO), where each unit or layer is strategically programmed to exhibit specific energy barriers and deployment sequences. The term “hierarchical” denotes organization at multiple levels, such as arrangements of individual CKOs into multi-layered networks with designed mechanical responses, typically for deployable systems, morphing materials, and robotic actuation. These assemblies leverage controlled energy landscapes at the unit and assembly levels to achieve programmable, sequential, and robust mechanical behavior.

1. Physics-Informed Neural Network (PINN) Framework for CKO

Hierarchical CKO assemblies are designed and analyzed using a physics-informed neural network (PINN) approach that enables both forward prediction and inverse design of the mechanical energy landscape, circumventing the need for pre-collected training data. The PINN architecture is a fully connected feedforward neural network with two hidden layers of 128 tanh-activated neurons each. Its input includes vertical deformation height ($h$) and a set of geometric parameters (such as crease lengths $a, b, c$, internal angle $\beta$, and polygon side number $n$); the output is the deformation-dependent rotation field $\varphi(h)$.

Key to the method is embedding the underlying physics into the loss function:

• The equilibrium (zero-torque) condition $\partial U / \partial \varphi = 0$ is directly enforced, where $U(h, \varphi)$ represents the mechanical energy.
• An Euler–Lagrange regularization term penalizes unphysical discontinuities in the predicted rotation profile, promoting smooth, physically consistent solutions:

$$L_{EL} = \mathrm{mean}\left[ \left( \frac{\partial U}{\partial \varphi} - 2A_\mathrm{reg} \frac{\partial^2 \varphi}{\partial h^2} \right)^2 \right]$$

• Predicted rotation angles are constrained to physically admissible intervals using a sigmoid-based transformation.

This formulation enables rapid, data-free learning of the equilibrium paths and energy landscapes for arbitrarily specified CKO geometries.

2. Structure and Programming of Hierarchical CKO Assemblies

A hierarchical CKO assembly consists of multiple CKO layers stacked (or otherwise spatially arranged), with each layer engineered to possess a distinct energy barrier—defined as the mechanical energy difference separating stable configurations—thereby enabling controlled, sequential deployment or actuation. The central design rationale is to assign each CKO layer a target barrier magnitude ($E_{\mathrm{barrier}}$), with the sequencing of deployment encoded in the ordering of these barrier heights.

For instance, in a three-layer stack, the lowest barrier is assigned to the bottom unit, while progressively higher barriers are given to the intermediate and top units (e.g., $0.05$ mJ, $0.08$ mJ, and $0.11$ mJ). Upon global compression or actuation, the system’s energy input is preferentially absorbed in overcoming the series of barriers, ensuring the layers deploy in a prescribed layer-by-layer manner.

The geometric and mechanical parameters of each CKO unit (creases and internal angles) are jointly optimized using differentiable mappings (e.g., logarithmic or sigmoid functions) within the inverse design routine to guarantee that barrier heights and deployment order precisely match design specifications.

3. Energy Landscape Modeling and Inverse Design Methodology

Each CKO unit is modeled as an elastic truss structure, with the total energy given by the sum of stretching energies stored in mountain and valley creases:

$$U(h, \varphi) = \sum_{\mathrm{mountain}} k_m [L_m(h, \varphi) - L_{m,0}]^2 + \sum_{\mathrm{valley}} k_y [L_y(h, \varphi) - L_{y,0}]^2$$

where $k_m$ and $k_y$ are stiffness coefficients, and $L_{m,0}$, $L_{y,0}$ are rest lengths.

The PINN is trained to fit the two-dimensional energy landscape $U(h, \varphi)$ and its equilibrium path (where $\partial U / \partial \varphi = 0$). For inverse design, loss terms include:

• Target matching loss $L_{\mathrm{target}}$, enforcing that the PINN-predicted $U(h)$ matches a desired target energy profile;
• Physical equilibrium loss $L_{\mathrm{phys}}$, ensuring mechanical torque balance;
• Euler–Lagrange regularization $L_{EL}$ for smoothness and variational consistency.

In bistable programming scenarios, the designer specifies only two stable heights and a desired barrier magnitude; the model infers all other parameters and the complete energy profile.

4. Validation via Numerical and Experimental Studies

The PINN framework’s forward and inverse design predictions are validated against both finite element analysis (FEA) and experimental physical prototypes:

• Forward predictions yield $R^2 > 0.93$ and root-mean-square errors on the order of $10^{-3}$ when compared to FEA.
• In multi-layer assemblies, experimental force–displacement and energy–displacement responses replicate the programmed deployment sequence; measured barrier heights follow the designed progression with minor quantitative deviations attributable to effects not captured by the elastic truss abstraction (such as non-ideal dissipation and complex local deformations).

Examples include cylindrical, conical, and zigzag multi-layered CKO assemblies, each of which demonstrates hierarchical, sequential deployment as the global system is compressed.

5. Implications for the Design, Modeling, and Assembly of Multiscale Structures

Programming energy barriers in hierarchical CKO assemblies enables deployment behavior not attainable by monolithic or non-hierarchically organized origami systems. Sequential deployment mitigates disruptive snap-through cascades, enhances controllability, and permits robust adaptation to environmental loads or mixed actuation strategies.

Potential applications include:

• Deployable aerospace systems (e.g., booms, reflectors, morphing skins) requiring controlled, multi-stage deployment.
• Soft robotic actuators exploiting programmable snap-through and force-limiting mechanisms.
• Impact-mitigation and biomedical devices where graded deployment is critical.

Hierarchical programming reflects a broader principle in self-assembly—mirroring thermodynamic and kinetic considerations from molecular systems—where direct, sequential assembly or actuation can outperform pathways involving larger, composite intermediates due to avoidance of kinetic traps and nonproductive aggregation (Haxton et al., 2012).

6. Figures and Quantitative Highlights

Key figures illustrate geometric modeling, energy landscapes, and PINN architecture:

• CKO unit geometry with labeled creases and rotational degrees of freedom.
• Sample $U(h, \varphi)$ energy landscape plots.
• Schematic diagrams of PINN input/output and loss function composition.
• Comparison of designed versus experimental energy barriers in multi-layer assemblies, demonstrating fidelity to prescribed deployment behavior.

A summary table of core numerical results follows:

Feature	Metric/Observation	Validation
Forward PINN prediction accuracy	$R^2 > 0.93$; RMSE $\sim10^{-3}$	FEA comparison
Inverse design target matching	Prescribed barriers (e.g., 0.05, 0.08, 0.11 mJ) achieved	Experiment/FEA
Sequential deployment in assembly	Layers fold in programmed order	Prototype tests

7. Broader Context: Self-Assembly, Hierarchical Pathways, and Design Rules

The hierarchical assembly paradigm in CKO reflects general principles in the design of multiscale materials. Direct, sequential pathways—in both molecular self-assembly and mechanical origami—are often preferred to avoid kinetic traps, unproductive intermediate phases, and aggregation. In thermodynamic terms, the optimal design ensures a moderate free energy gap ($\Delta G \sim k_B T$) between the target structure and most stable fluid phase, with dense competing or incomplete phases lying well above this gap (Haxton et al., 2012). Analogously, hierarchical CKO assemblies exploit layer-by-layer energy barrier programming to exert fine control over the system’s kinetic and mechanical pathway, favoring robust functional deployment.