Rigid Origami

Rigid origami is a unique subset of origami, where continuous folding motions are generated by its facets rotating around creases without being stretched or bent. This key feature has led to great potential in engineering applications, such as foldable structures that consist of rigid materials.

1 Kinematics of Rigid Origami Pattern

Based on the kinematic equivalence between zero-thickness rigid origami and spherical mechanisms, multi-vertex origami patterns are equivalent to assemblies of spherical mechanisms. Through systematic research on kinematic compatibility conditions, a universal kinematic model for rigid origami was established. A one-degree-of-freedom (DOF) rigid origami synthesis technique based on symmetry assumptions was proposed. Taking the assembly of spherical 4R mechanisms as a basis, three one-DOF combination forms, including dual-plane symmetry, plane symmetry, and rotational symmetry, were constructed. By adjusting the geometric parameters of mechanisms according to the symmetry of trigonometric functions, various one-DOF rigid origami patterns were generated, including a novel helical pattern rigid origami. These studies provide a theoretical foundation for the kinematic analysis of complex origami patterns and the innovative design of rigid origami patterns.

Kinematics model of a
four-crease vertex

Kinematics model of multi-vertex rigid origami

A novel helical pattern
  • Yan Chen, Weilin Lv, Rui Peng, Guowu Wei*. Mobile assemblies of four-spherical-4R-integrated linkages and the associated four-crease-integrated rigid origami patterns. Mechanism and Machine Theory, 2019, 142: 103613.
    (https://doi.org/10.1016/j.mechmachtheory.2019.103613)

2 Mountain-Valley Crease Assignments of Origami Patterns

Systematic methods for analyzing and designing the arrangement of mountain-valley creases in origami patterns were developed. Classic origami patterns, such as the triangle twist, double corrugated, and square twist patterns were studied by analyzing kinematic compatibility conditions under different mountain-valley crease assignments. All possible arrangements that allow rigid motion, i.e., rigid origami, were identified. For tessellations of four-crease vertices, an algorithm was developed that combines graphic representation and motion compatibility conditions. This algorithm facilitates the rapid identification of feasible mountain-valley crease assignments and provides a strong theoretical foundation for the innovative design of rigid origami. These studies are beneficial for engineering applications of origami in deployable structures, metamaterials, and origami robots.

Several rigid-foldable mountain-valley crease assignments for double
corrugated origami

Different rigid-foldable mountain-valley crease assignments for a tessellation of four-crease vertices
  • Huijuan Feng, Rui Peng, Jiayao Ma, Yan Chen*. Rigid foldability of generalized triangle twist origami pattern and its derived 6R linkages. Trans. ASME. Journal of Mechanisms and Robotics, 2018, 10(5): 051003.
    (https://doi.org/10.1115/1.4040439)
  • Rui Peng, Jiayao Ma, Yan Chen*. The Effect of Mountain-Valley Folds on the Rigid Foldability of Double Corrugated Pattern. Mechanism and Machine Theory, 2018, 128: 461–474.
    (https://doi.org/10.1016/j.mechmachtheory.2018.06.012)
  • Huijuan Feng, Rui Peng, Shixi Zang, Jiayao Ma, Yan Chen*. Rigid foldability and mountain-valley crease assignments of square-twist origami pattern. Mechanism and Machine Theory, 2020, 152: 103947.
    (https://doi.org/10.1016/j.mechmachtheory.2020.103947)
  • Weiqi Liu, Song Cao, Yan Chen*. Mountain-valley crease reconfiguration of 4-crease origami vertices and tessellations. International Journal of Mechanical Sciences, 2024, 273: 109224.
    (https://doi.org/10.1016/j.ijmecsci.2024.109224)

3 Vertex-Splitting

To address the complex motion and challenging control of multi-DOF rigid origami, the “vertex-splitting” technique was proposed to reduce the DOF of origami patterns. Taking diamond origami as an example, a single vertex can be split in horizontal, vertical, or both horizontal and vertical directions. When applied to multi-DOF multi-vertex diamond origami, “vertex-splitting” yields a total of 42 one-DOF origami patterns. Among these, there are new one-DOF origami patterns composed of both four-crease and six-crease vertices, successfully achieving a reduction in DOF. This technique can also be applied to other multi-DOF origami patterns, providing a novel approach for the innovative design of one-DOF rigid origami and facilitating single-actuator control.

The vertex-splitting results
of a diamond origami vertex

A one-DOF vertex-splitting result of a multi-vertex diamond origami
  • Xiao Zhang, Yan Chen*. Vertex-Splitting on a Diamond Origami Pattern. Trans. ASME. Journal of Mechanisms and Robotics, 2019, 11(3): 031014.
    (https://doi.org/10.1115/1.4043214)

4 Rigid Origami Morphing Structures with Multiple States

By integrating the theory of spatial mechanisms with rigid origami, a novel approach based on crease superposition, crease activation, and dormancy was proposed to design morphing structures with multiple states. Utilizing crease superposition, a rigid origami pattern with a large deployed/folded ratio and capable of continuous motion was created. Furthermore, control of the folding sequence was realized through the mountain-valley transformation of creases. Based on an origami pattern consisting of four-crease and six-crease vertices, the activation and dormancy of creases were introduced to construct a reconfigurable origami structure. By alternately activating and deactivating creases, geometric reconfiguration was realized. These studies are of great significance for the innovative design of multiple-state deployable structures, intelligent robots, and programmable metamaterials.

Two rigid origami morphing structures

  • Xiang Liu, Joseph M. Gattas, Yan Chen*. One-DOF Superimposed Rigid Origami with Multiple States. Scientific Reports, 2016, 6: 36883.
    (https://doi.org/10.1038/srep36883)
  • Chunlong Wang, Hongwei Guo*, Rongqiang Liu, Zongquan Deng, Yan Chen, Zhong You*. Reconfigurable Origami-inspired Multistable Metamorphous Structures. Science Advances, 2024, 10(22): eadk8662.
    (https://doi.org/10.1126/sciadv.adk8662)

5 Tubular Rigid Origami

Tubular origami structures are equivalent to spatial assemblies of spherical mechanisms. By solving highly nonlinear inverse kinematics, the crease distribution rules for tubular rigid origami were obtained, allowing the design of a one-DOF flat-foldable tubular origami structure with plane symmetry. Furthermore, new types of one-DOF rigid-foldable tubular structures were constructed by either employing additional facets onto each modular unit or combining two one-DOF tubes to a new configuration. These methods were applied not only to multilayered vertical tubes but also to radially assembled arc profiles, offering design possibilities for a variety of engineering applications, including deployable structures, metamaterials, and origami-based robotics. We invented an origami tent based on a one-DOF rigid-foldable tubular origami, which is lighter, cheaper, and more environmentally friendly compared to traditional tent designs. Our design won the department award of earthquake reconstruction in the competition of Japan technology and business plan in 2012, and was invited to be exhibited at the 2024 Sustainable Social Value Innovation Summit.

     

The kinematics model of tubular origami and combination of tudes

Tubular rigid origami
structures with a kite
or pentagon cross-section

Origami tent
  • Sicong Liu, Weilin Lv, Yan Chen*, Guoxing Lu. Deployable prismatic structures with rigid origami patterns. Trans. ASME. Journal of Mechanisms and Robotics, 2015, 8(3): 031002.
    (https://doi.org/10.1115/1.4031953)
  • Yan Chen, Weilin Lv, Junlan Li, Zhong You*. An extended family of rigidly foldable origami tubes. Trans. ASME. Journal of Mechanisms and Robotics, 2017, 9(2): 021002.
    (https://doi.org/10.1115/1.4035559)

6 Rigid and Flat-Foldable Origami Polyhedrons

To address the challenge that closed polyhedrons cannot be rigidly folded, several techniques that introduce additional creases and slits were proposed to create origami polyhedrons. By symmetrically adding diagonal creases and slits to the surfaces of a rhombic dodecahedron and a tetrahedron, and by performing crease projection and removal operations, various one-DOF rigid and flat-foldable origami polyhedrons were created. To achieve parallel motion between the upper and lower square facets while preserving their structural integrity, a systematic technique based on 3D straight-skeleton and mechanism equivalence was proposed to arrange the creases and slits. By establishing the relationship between DOF and slit arrangement, various one-DOF origami cubes were created. These studies not only expand the family of origami polyhedrons and their design theory but also demonstrate promising engineering applications in deployable modular cabins and programmable mechanical metamaterials.

                   

Origami rhombic dodecahedron        Origami tetrahedron              Origami cube