Unified Origami Kinematics via Cosheaf Homology

TL;DR

The paper tackles the challenge of analyzing rigid origami kinematics across multiple modeling paradigms (hinge, spatial, and truss) on complex topologies. It introduces a cosheaf-homology framework in which a network of exact sequences and connecting morphisms links these models, enabling global analysis without unfolding the origami. Central contributions include explicit maps, notably the connecting morphism $\vartheta: H_2\mathcal{S} \to H_1\mathcal{H}$ with $\vartheta^+$, and the isomorphism $\eta: H_2\mathcal{S} \to \ker{\bf M}'$ that connect spatial and truss descriptions; together they yield $\eta\vartheta^+|_{\ker\iota_*}$ as an overall isomorphism between hinge and truss solutions up to global $SE(3)$ motion. This framework enables simultaneous, global kinematic analysis of origami structures and points toward homological forward-kinematics algorithms and optimization strategies for robotic origami and related systems.

Abstract

We establish a novel local-global framework for analyzing rigid origami mechanics through cosheaf homology, proving the equivalence of truss and hinge constraint systems via an induced linear isomorphism. This approach applies to origami surfaces of various topologies, including sheets, spheres, and tori. By leveraging connecting homomorphisms from homological algebra, we link angular and spatial velocities in a novel way. Unlike traditional methods that simplify complex closed-chain systems to re-constrained tree topologies, our homological techniques enable simultaneous analysis of the entire system. This unified framework opens new avenues for homological algorithms and optimization strategies in robotic origami and beyond.

Figures (7)

• Figure 1: Kinematic data assignments of different origami kinematic models are pictured over a sample face, edge, and vertex. The hinge and spatial models have "complementary" cellular geometric data assignments in the sense that one has data where the other lacks it.
• Figure 2: An origami serial chain. Spatial velocities, angular velocities, and rigid body operators are pictured (left) . The dual graph (right) is directed along the recursive equation \ref{['eq:recurrence']}. This graph is acyclic, indicating a open-chain system.
• Figure 3: The degrees of freedom and constraints of the hinge model (left) and truss model (right) are displayed. The rotational velocities sum to zero around each vertex (left) and the length of the edge changes by zero (right). The data assignments of Figure \ref{['fig:overview']} are divided into degrees of freedom and constraints.
• Figure 4: The analogues to Figure \ref{['fig:constraints']} for the spatial model $\mathcal{S}$ (left) and rigid model $\mathcal{B}$ (right) are pictured. [Left] Faces have six degrees of freedom and are subject to three translational and two rotational constraints over edges where only hinged rotation is permitted. Over the pictured system the homology space $H_2 \mathcal{S}$ is seven-dimensional, the kernel of the associated size $5 \times 12$ boundary matrix $\partial_\mathcal{S}$. Six generators of $H_2 \mathcal{S}$ characterize global degrees of freedom while the last generator is the hinge action. [Right] As a unified rigid body, edges have six degrees of constraints locking adjacent faces in place relative to each other.
• Figure 5: The connecting homomorphism $\vartheta$ sends solutions of the spatial model $\nu\in H_2 \mathcal{S}$ to solutions of the hinge model $H_1 \mathcal{H}$ (consisting of angular velocities $\dot{\theta}$). Global velocity vectors at all faces (thought of as elements of $H_2 \mathcal{B}$) do not bend joints, so $\vartheta$ sends these to zero.

Theorems & Definitions (26)

• Theorem : \ref{['thm:holes']}
• Theorem : \ref{['thm:eta_hom']}
• Corollary 1
• Definition 2: Origami
• Definition 3: Hinge cosheaf
• Lemma 4
• proof
• Definition 5: Spatial cosheaf
• Lemma 6
• proof