## Abstract

In rigid origami, the complex folding motion arises from the rotation of strictly rigid faces around crease lines that represent perfect revolute joints. The rigid folding motion of an origami crease pattern is collectively determined by the kinematics of its individual vertices. Establishing a kinematic model and determining the conditions for the rigid foldability of a single vertex is thus important to exploit rigid origami in engineering design tasks. Today, there exists neither an efficient kinematic model to determine the unknown dihedral angles nor an intrinsic condition for the rigid foldability of arbitrarily complex vertices of degree n. In this paper, we present the principle of three units (PTU) that provides an efficient approach to modeling the kinematics of single degree-n vertices. The PTU is based on the notion that the kinematics of a vertex is determined by the behavior of a single underlying spherical triangle. The condition for the existence of this triangle leads to the condition for the rigid and flat foldability of degree-n vertices. These findings are transferred from single vertices to crease patterns, resulting in a simple rule to generate kinematically determinate crease patterns that can be designed to fold rigidly. Finally, we discuss the limitations of the PTU with respect to the global rigid foldability of a crease pattern.

## References

1.
Waitukaitis
,
S.
, and
van Hecke
,
M.
,
2016
, “
Origami Building Blocks: Generic and Special Four-Vertices
,”
Phys. Rev. E
,
93
(
2
), p.
023003
. 10.1103/PhysRevE.93.023003
2.
Belcastro
,
S.-M.
, and
Hull
,
T. C.
,
2002
, “
A Mathematical Model for Non-Flat Origami
,”
Origami 3: Proceedings of the 3rd International Meeting of Origami Science, Mathematics, and Education
,
Asilomar, CA
,
Mar. 9–11, 2001
, pp.
39
51
.
3.
Balkcom
,
D. J.
, and
Mason
,
M. T.
,
2008
, “
Robotic Origami Folding
,”
Int. J. Rob. Res.
,
27
(
5
), pp.
613
627
. 10.1177/0278364908090235
4.
Miura
,
K.
,
1989
, “A Note on Intrinsic Geometry of Origami,”
Research of Pattern Formation
,
R.
Takaki
, ed.,
KTK Scientific Publishers
,
Tokyo, Japan
, pp.
91
102
.
5.
He
,
Z.
, and
Guest
,
S. D.
,
2018
, “
On Rigid Origami II: Quadrilateral Creased Papers
,” preprint arXiv:1804.06483, Accessed Apr. 17, 2018.
6.
Huffman
,
D.
,
1976
, “
Curvature and Creases: A Primer on Paper
,”
IEEE Trans. Comput.
,
C-25
(
10
), pp.
1010
1019
. 10.1109/TC.1976.1674542
7.
Lang
,
R. J.
,
Magleby
,
S.
, and
Howell
,
L.
,
2016
, “
Single Degree-of-Freedom Rigidly Foldable Cut Origami Flashers
,”
ASME J. Mech. Rob.
,
8
(
3
), p.
031005
. 10.1115/1.4032102
8.
Zimmermann
,
L.
,
Shea
,
K.
, and
Stanković
,
T.
,
2019
, “
Rigid and Flat Foldability of a Degree-Four Vertex in Origami
,”
ASME J. Mech. Rob.
,
12
(
1
), p.
011004
. 10.1115/1.4044737
9.
Justin
,
J.
,
1986
, “
Mathematics of Origami, Part 9
,”
British Origami
,
118
, pp.
28
30
.
10.
Kawasaki
,
T.
,
1991
, “
On the Relation Between Mountain-Creases and Valley-Creases of a Flat Origami
,”
Origami 1: Proceedings of the 1st International Meeting of Origami Science and Technology
,
Ferrara, Italy
,
Dec. 6–7, 1989
, pp.
229
237
.
11.
Tachi
,
T.
,
2009
, “
,”
J. Int. Assoc. Shell Spatial Struct.
,
50
(
3
), pp.
173
179
.
12.
Tachi
,
T.
,
2010
, “
Geometric Considerations for the Design of Rigid Origami Structures
,”
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium, Shanghai
,
China
,
Nov. 8–12
, Vol.
12
(
10
), pp.
458
460
.
13.
Evans
,
T. A.
,
Lang
,
R. J.
,
Magleby
,
S.
, and
Howell
,
L.
,
2015
, “
Rigidly Foldable Origami Twists
,”
Origami 6: Proceedings of the 6th International Meeting of Origami Mathematics
,
Tokyo, Japan
,
Aug. 10–13, 2014
,
Science and Education
, pp.
119
130
.
14.
Evans
,
A. A.
,
Silverberg
,
J. L.
, and
Santangelo
,
C. D.
,
2015
, “
Lattice Mechanics of Origami Tessellations
,”
Phys. Rev. E
,
92
(
1
), p.
013205
. 10.1103/PhysRevE.92.013205
15.
Cai
,
J.
,
Deng
,
X.
,
Zhang
,
Y.
,
Feng
,
J.
, and
Zhou
,
Y.
,
2016
, “
Folding Behavior of a Foldable Prismatic Mast With Kresling Origami Pattern
,”
ASME J. Mech. Robot.
,
8
(
3
), p.
031004
. 10.1115/1.4032098
16.
Cai
,
J.
,
Deng
,
X.
,
Xu
,
Y.
, and
Feng
,
J.
,
2016
, “
Motion Analysis of a Foldable Barrel Vault Based on Regular and Irregular Yoshimura Origami
,”
ASME J. Mech. Robot.
,
8
(
2
), p.
021017
. 10.1115/1.4031658
17.
Feng
,
H.
,
Peng
,
R.
,
Ma
,
J.
, and
Chen
,
Y.
,
2018
, “
Rigid Foldability of Generalized Triangle Twist Origami Pattern and Its Derived 6r Linkages
,”
ASME J. Mech. Robot.
,
10
(
5
), p.
051003
. 10.1115/1.4040439
18.
Ramsay
,
A.
, and
Richtmyer
,
R. D.
,
2013
,
Introduction to Hyperbolic Geometry
,
,
NY
, p.
17
.
19.
Chen
,
B. G. G.
, and
Santangelo
,
C. D.
,
2018
, “
Branches of Triangulated Origami Near the Unfolded State
,”
Phys. Rev. X
,
8
(
1
), p.
011034
. 10.1103/PhysRevX.8.011034
20.
Kamrava
,
S.
,
,
D.
,
Felton
,
S. M.
, and
Vaziri
,
A.
,
2018
, “
Programmable Origami Strings
,”
,
3
(
3
), p.
1700276
21.
Lang
,
R. J.
, and
Howell
,
L.
,
2018
, “
Rigidly Foldable Quadrilateral Meshes From Angle Arrays
,”
ASME J. Mech. Robot.
,
10
(
2
), p.
021004
. 10.1115/1.4038972
22.
Bowen
,
L. A.
,
Grames
,
C. L.
,
Magleby
,
S. P.
,
Howell
,
L. L.
, and
Lang
,
R. J.
,
2013
, “
A Classification of Action Origami as Systems of Spherical Mechanisms
,”
ASME J. Mech. Design
,
135
(
11
), p.
111008
. 10.1115/1.4025379
23.
Streinu
,
I.
, and
Whiteley
,
W
.,
2004
, “Single-vertex Origami and Spherical Expansive Motions,”
Japanese Conference on Discrete and Computational Geometry
,
J.
Akiyama
,
M.
Kano
, and
X.
Tan
eds.,
Springer
,
Berlin, Heidelberg
, pp.
161
173
.
24.
Cai
,
J.
,
Zhang
,
Q.
,
Feng
,
J.
, and
Xu
,
Y.
,
2019
, “
Modeling and Kinematic Path Selection of Retractable Kirigami Roof Structures
,”
Computer-Aided Civil Infrastruct. Eng.
,
34
(
4
), pp.
352
363
. 10.1111/mice.12418
25.
Savage
,
M.
, and
Strickland Hall
,
A.
,
1970
, “
Unique Descriptions of All Spherical Four-Bar Linkages
,”
J. Eng. Ind.
,
92
(
3
), pp.
559
563
. 10.1115/1.3427812
26.
Abel
,
Z.
,
Cantarella
,
J.
,
Demaine
,
E. D.
,
Eppstein
,
D.
,
Hull
,
T. C.
,
Ku
,
J. S.
,
Lang
,
R. J.
, and
Tachi
,
T.
,
2016
, “
Rigid Origami Vertices: Conditions and Forcing Sets
,”
J. Comput. Geom.
,
7
(
1
), pp.
171
184
.
27.
Liu
,
B.
,
Silverberg
,
J. L.
,
Evans
,
A. A.
,
Santangelo
,
C. D.
,
Lang
,
R. J.
,
Hull
,
T. C.
, and
Cohen
,
I.
,
2018
, “
Topological Kinematics of Origami Metamaterials
,”
Nat. Phys.
,
14
(
8
), pp.
811
815
. 10.1038/s41567-018-0150-8
28.
Zimmermann
,
L.
,
Shea
,
K.
, and
Stanković
,
T.
,
2018
, “
Origami Sensitivity—On the Influence of Vertex Geometry
,”
Origami 7: Proceedings of the 7th International Meeting of Origami Mathematics
,
Oxford, UK
,
Sept. 5–7
,
Science and Education
, pp.
1087
1102
.
29.
Tachi
,
T
.,
2010
, “Freeform Rigid-Foldable Structure Using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh,”
,
C.
Ceccato
,
L.
Hesselgren
,
M.
Pauly
,
H.
Pottmann
, and
J.
Wallner
, eds.,
Springer
,
Vienna
, pp.
87
102
.
30.
Miura
,
K.
,
1985
, “
Method of Packaging and Deployment of Large Membranes in Space
,”
Inst. Space Astronaut. Sci. Rep.
,
618
, pp.
1
9
.
31.
Edmondson
,
B. J.
,
Bowen
,
L. A.
,
Grames
,
C. L.
,
Magleby
,
S. P.
,
Howell
,
L. L.
, and
Bateman
,
T. C.
,
2013
, “
Oriceps: Origami-Inspired Forceps
,”
ASME 2013 Conference on Smart Materials, Adaptive Structures and Intelligent Systems
,
Snowbird, UT
,
Sept. 16–18
, Vol.
1
,
V001T01A027
.