A Unified Framework for Kinematic Simulation of Rigid Foldable Structures

TL;DR

This work addresses the challenge of unified kinematic analysis for rigid-foldable structures by introducing a data-schema–driven, graph-based framework that automatically identifies independent loop closures via a minimum cycle basis, assigns consistent screw axes, and constructs a velocity-level Pfaffian constraint matrix $A(\theta)$ such that $A(\theta)\dot{\theta}=0$. The approach integrates geometric primitives with screw theory to model coupled rotational and translational loop closures in a coordinate-free form, enabling scalable analysis across origami, kirigami, thick-panel, and multi-sheet configurations. Key contributions include the automated data representation, automatic loop extraction, consistent screw alignment, and the construction of loop Jacobians for a compact, extensible kinematic model, demonstrated on large-scale patterns with open-source implementation. The framework provides a robust, geometry-driven path to compute feasible folding trajectories and visualizations, facilitating design exploration and dynamic extensions in rigid origami-inspired systems.

Abstract

Origami-inspired structures with rigid panels now span thick, kirigami, and multi-sheet realizations, making unified kinematic analysis essential. Yet a general method that consolidates their loop constraints has been lacking. We present an automated approach that generates the Pfaffian constraint matrix for arbitrary rigid foldable structures (RFS). From a minimally extended data schema, the tool constructs the facet-hinge graph, extracts a minimum cycle basis that captures all constraints, and assembles a velocity-level constraint matrix via screw theory that encodes coupled rotation and translation loop closure. The framework computes and visualizes deploy and fold motions across diverse RFS while eliminating tedious and error-prone constraint calculations.

Figures (12)

• Figure 1: Overview of the proposed computational pipeline for constructing the Pfaffian constraint matrix of an RFS.
• Figure 2: Examples of sheet-wise connections and their graph representation in the proposed data schema. (A) Hinging-type connection introducing a rotational degree of freedom in a stacked Miura origami (SMO) unit cell. (B) Soldering-type connection constraining relative motion in a Tachi--Miura polyhedron (TMP) unit cell. In the SMO case, hinging adds graph edges, whereas in the TMP case, soldering merges nodes into a single one, resulting in distinct graph representations.
• Figure 3: Closure loops in patterns and cycles in graphs are in one-to-one correspondence (left: pattern, right: graph). (A) Facet–hinge graph representation of a rigid-foldable kirigami pattern. (B) A complete and minimum set of closure loops obtained via the minimum cycle basis. (C) A complete but non-minimum loop set that contains redundant cycles.
• Figure 4: Schematic illustration of screw-axis orientation consistency and its influence on motion continuity. (A) Consistent facet orientation yields a uniform hinge motion. (B) Reversed orientation causes conflicting screw directions and motion discontinuity.
• Figure 5: (A) Screw assignment and traversal direction of a single minimal closure loop. (B) Configuration after folding motion, where the loop closure constraint is satisfied.