Mono-Actuated GeoGami Systems

• Mono-Actuated GeoGami is a system where a single input triggers a precise transformation from a 2D configuration to a pre-programmed 3D form.
• Mechanical modeling combines crease penalties and bending energy to achieve multistability and robust geometric transitions along a one-dimensional actuation path.
• Algorithmic design and kinematic coupling enable controlled deployment in soft robotic platforms and adaptive metamaterials with minimal actuator complexity.

Mono-Actuated GeoGami refers to geometric origami and kirigami systems engineered so that a single actuation input (mechanical, thermal, pneumatic, or other) robustly drives the transformation of the entire structure from its initial (often planar or stowed) configuration to a pre-programmed, three-dimensional deployed state. These systems exploit geometric frustration, kinematic coupling, material metric programming, and configuration space engineering to yield complex and functional shape changes, programmable motion, or load-bearing morphologies, while minimizing actuator count and control complexity.

1. Geometric Foundations of Mono-Actuated Origami

Mono-actuated GeoGami designs are rooted in the geometry of rigid, developable, or isometric deformations. In structures based on curved crease origami, the fundamental relationship between the dihedral angle $\theta$ at the crease and the local curvature $\kappa$ is

$\sin(\theta/2) = 1/\kappa$

as established for circularly creased annuli (Dias et al., 2012). Isometric conditions enforce that the surfaces away from the crease are developable, and the three-dimensional shape is determined up to reparametrization by the curvature and torsion of the crease, constrained by generator angles and the fold width. For planar periodic origami such as Miura-ori, the deformation is uniquely controlled by a set of geometric parameters (parallelogram angle $\alpha$ and folding angle $\theta$), ensuring a single degree of freedom (DOF) rigid motion over the configuration space (Wei et al., 2012).

In both contexts, the geometry is instantiated in the design phase so that one input parameter—typically a global fold angle, curvature, or actuation level—commutes the entire structure between structurally distinct (metastable) states, inherently limiting the actuation space to one-dimensional curves in configuration space.

2. Mechanical Modeling and Energy Landscapes

The mechanics of mono-actuated GeoGami systems are quantified by combining the bending energy of developable sheets and the potential energy stored or dissipated in creases. For curved crease origami, the total elastic energy involves both the mean curvature energy of the panels (integrated over developables) and a crease penalty function: $$E_c \propto \int [\cos(\theta/2) - \cos(\theta_0/2)]^2 ds$$ where $\theta_0$ is the rest angle imposed by material or pre-strain (Dias et al., 2012). In regimes of high crease stiffness, shape selection is dominated by sheet bending, resulting in nearly constant curvature and oscillatory torsion, while reduced crease stiffness allows for symmetry breaking with oscillatory curvature and torsion.

Bifurcation and multistability are critical design features. For example, in origami bellows, the analytic rigid-face model predicts two accessible rigid states, with deployability governed by the bistability gap $\Delta\psi$ between these states (Reid et al., 2016). In actively actuated systems, such as non-Euclidean origami with prescribed metric programming, energetic cost and, consequently, actuation effort are tied to how closely the geometric design permits isometric transitions between states (Dudte et al., 2018, Wang et al., 2 Jun 2025). In non-Euclidean 4-vertices, branch splitting lifts degeneracy and creates robust tristable or multistable behavior which suppresses misfolding (Waitukaitis et al., 2019).

3. Kinematic Coupling, Configuration Space, and Control

A distinctive feature of mono-actuated GeoGami is the deliberate engineering of configuration space to restrict the accessible degrees of freedom. For example, in systems built from modified Miura-ori vertices of degree 6, local parameters $(t_1, t_2, t_3)$ span a higher-dimensional configuration space, but global constraints and inter-vertex coupling effectively collapse motion onto a low-dimensional (often one-dimensional) manifold (Liu et al., 2017). Topological tailoring—such as diode-like asymmetry or carefully designed bifurcations in the configuration space—renders specific folding paths robust to fabrication error and parasitic compliance, and supports mechanical diodes or sequential snap-through transitions.

Distributed actuation via spontaneous curvature exploits kinematic cooperativity: a global field (e.g., thermal, hygromorphic, or nematic in LCEs) programs metric or curvature changes into the origami panels, with geometric constraints and mechanical coupling synchronizing the folding pathway to produce a robust collective response—even with a single actuation parameter (DeSimone et al., 24 Dec 2024, Wang et al., 2 Jun 2025).

4. Algorithmic and Computational Strategies

Achieving mono-actuated behavior in complex origami/kirigami requires advanced computational tools for both forward and inverse design. For planar pixel-matrix designs, the search for valid crease patterns supporting mono-actuation is reduced from an exponential space ($2^{N^2}$ for an $N \times N$ sheet) to enumeration over grid-graph spanning trees using Kirchhoff's matrix–tree theorem, integrating combinatorial topology and graph theory (Dureisseix, 2015).

For simulation and actuation control, Lagrange multiplier methods efficiently update fold angles subject to geometric compatibility (loop closure constraints), and simplify to one-dimensional updates for single-DOF (mono-actuated) systems (Hu et al., 2020). For isometric immersions in kirigami, the design of cut patterns determines polygonal geodesic pathways for force transmission, and the mapping of these paths into energetically favorable flat-folded (piecewise affine) configurations—leveraging the interplay between planar geometry and spatial deployment (Han et al., 2021).

Optimization algorithms (genetic, surrogate, greedy best-first search) have become central for actuation path planning in modular bistable origami robots capable of multimodal deployment or spatially targeted reach with a single input (Forte et al., 2021).

5. Extensions to Non-Euclidean and Actively-Programmed Origami

Active origami and kirigami structures integrating metric programming—via director fields in LCE sheets or actuator-induced spatial curvature—realize mono-actuated GeoGami at higher complexity. Non-Euclidean origami vertex theory, both in four-vertex and general $N$-vertex settings, provides analytical expressions relating the director-induced metric changes to achievable folding angles and concentrated Gaussian curvature at vertices (Wang et al., 2 Jun 2025, Waitukaitis et al., 2019). Metric compatibility across interfaces is encoded in vector geometric relations and sector angle matching, while continuum approaches based on piecewise-constant nematic fields furnish designable deformation spaces (e.g., unique solutions for three-fold vertices; one-parameter manifolds for four-fold vertices).

Programming the actuation via spatially modulated director fields enables mono-actuated structures exhibiting defect-induced curvature localization, pop-through transitions, and global shape change with a single environmental signal (e.g., light, temperature).

6. Physical Realizations and Robotic Platforms

Mono-actuated GeoGami has matured into robust, soft-rigid robotic platforms that integrate compliant geometric skeletons, programmable origami surfaces, and centralized cyclic actuation (including cable-driven gearboxes or centralized pneumatic actuation). In these designs, underactuation is addressed by embedding compliance in both the skeleton and origami surface, balancing force–angle response through a series-parallel combination of flexure and surface mechanics (Webster et al., 25 Sep 2025). By time-multiplexing servo or cable pulls among corners of a ring skeleton, these robots achieve rolling, shape contraction, or translation with just a single actuator—verified by effective stiffness and center-of-mass shift models. Experimental prototypes demonstrate shape transformation, repeatable rolling, and environmental adaptability with minimal actuation complexity.

Similar principles govern origami crawlers constructed from a single rigid, degree-four vertex actuated by a pneumatic pouch: geometric tuning and nonlinear frictional interactions with the substrate allow the crawler to switch between straight gait and turning, controlled solely by the range of the folding angle during cyclic actuation (Farhadi et al., 28 Oct 2024).

7. Applications and Future Directions

Mono-actuated GeoGami enables compact deployable structures (e.g., bellows, stowable boxes, architectural panels), field-adaptable robots, metamaterials with tunable stiffness or energy absorption, and surfaces morphing to prescribed curvatures. In all these domains, mono-actuation ensures minimalism in the actuator array, energy efficiency, and tight integration between material geometry and function.

Key directions include:

• The further integration of non-Euclidean geometry and metric programming for adaptive, self-locking, and multistable responses.
• Algorithmic advances in the inverse design of actuation paths, including scale-independent tiling strategies, and optimization under fabrication or field constraints.
• Extension to higher-dimensional origami/surface origami constructions for the generation of programmable lattices and latticed actuators (Banerjee et al., 2021).
• Applications in soft robotics, adaptive architecture, and reconfigurable metamaterials predicated on the efficient translation of global actuation into controlled, robust shape change via geometric and mechanical design.

Mono-actuated GeoGami thus exemplifies the convergence of differential geometry, elasticity, combinatorics, and robotics, providing both a design framework and practical blueprint for translating minimal inputs into programmable, multi-scale mechanical functionalities.