Origami-type cartons have been widely used in packaging industry because of their versatility, but there is a lack of systematic approach to study their folding behavior, which is a key issue in designing packaging machines in packaging industry. This paper addresses the fundamental issue by taking the geometric design and material property into consideration, and develops mathematical models to predict the folding characteristics of origami cartons. Three representative types of cartons, including tray cartons, gable cartons, and crash-lock cartons were selected, and the static equilibrium of folding process was developed based on their kinematic models in the frame work of screw theory. Subsequently, folding experiments of both single crease and origami carton samples were conducted. Mathematical models of carton folding were obtained by aggregating single crease's folding characteristics into the static equilibrium, and they showed good agreements with experiment results. Furthermore, the mathematical models were validated with folding experiments of one complete food packaging carton, which shows the overall approach has potential value in predicting carton's folding behavior with different material properties and geometric designs.

## References

1.
,
T.
,
1980
, “
A Theory of Origami World
,”
Artif. Intell.
,
13
(
3
), pp.
279
311
.10.1016/0004-3702(80)90004-1
2.
Lang
,
R. J.
, and
Hull
,
T. C.
,
2005
, “
Origami Design Secrets: Mathematical Methods for an Ancient Art
,”
Math. Intell.
,
27
(
2
), pp.
92
95
.10.1007/BF02985811
3.
Dai
,
J. S.
, and
Caldwell
,
D.
,
2010
, “
Origami-Based Robotic Paper-and-Board Packaging for Food Industry
,”
Trends Food Sci. Technol.
,
21
(
3
), pp.
153
157
.10.1016/j.tifs.2009.10.007
4.
Dai
,
J. S.
, and
Jones
,
J.
,
1999
, “
Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds
,”
ASME J. Mech. Des.
,
121
(
3
), pp.
375
382
.10.1115/1.2829470
5.
Dai
,
J. S.
, and
Jones
,
J.
,
2002
, “
Kinematics and Mobility Analysis of Carton Folds in Packing Manipulation Based on the Mechanism Equivalent
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
216
(
10
), pp.
959
970
.10.1243/095440602760400931
6.
Leal
,
E.
, and
Dai
,
J. S.
,
2007
, “
From Origami to a New Class of Centralized 3-DOF Parallel Mechanisms
,”
Proceedings of ASME 31st Mechanisms and Robotics Conference
, Parts A and B, Las Vegas, NV, Sept. 4–7, ASME, New York, pp.
1183
1193
.
7.
Wei
,
G.
, and
Dai
,
J. S.
,
2009
, “
Geometry and Kinematic Analysis of an Origami-Evolved Mechanism Based on Artmimetics
,”
ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots
, ReMAR 2009, London, June 22–24, IEEE, New York, pp.
450
455
.
8.
Dai
,
J. S.
, and
Delun
,
W.
,
2007
, “
Geometric Analysis and Synthesis of the Metamorphic Robotic Hand
,”
ASME J. Mech. Des.
,
129
(
11
), pp.
1191
1197
.10.1115/1.2771576
9.
Cui
,
L.
, and
Dai
,
J. S.
,
2011
, “
Posture, Workspace, and Manipulability of the Metamorphic Multifingered Hand With an Articulated Palm
,”
ASME J. Mech. Rob.
,
3
(
2
), p. 021001.10.1115/1.4003414
10.
Lu
,
L.
, and
Akella
,
S.
,
2000
, “
Folding Cartons With Fixtures: A Motion Planning Approach
,”
IEEE Trans. Rob. Autom.
,
16
(
4
), pp.
346
356
.10.1109/70.864227
11.
Balkcom
,
D.
, and
Mason
,
M.
,
2004
, “
Introducing Robotic Origami Folding
,”
Proceedings
of IEEE International Conference on Robotics and Automation
, ICRA’04, Vol.
4
, IEEE, New York, pp.
3245
3250
.
12.
Balkcom
,
D.
, and
Mason
,
M.
,
2008
, “
Robotic Origami Folding
,”
Int. J. Robot. Res.
,
27
(
5
), pp.
613
627
.10.1177/0278364908090235
13.
Mullineux
,
G.
,
Feldman
,
J.
, and
Matthews
,
J.
,
2010
, “
Using Constraints at the Conceptual Stage of the Design of Carton Erection
,”
Mech. Mach. Theory
,
45
(
12
), pp.
1897
1908
.10.1016/j.mechmachtheory.2010.08.001
14.
Mullineux
,
G.
, and
Matthews
,
J.
,
2010
, “
Constraint-Based Simulation of Carton Folding Operations
,”
Comput.-Aided Des.
,
42
(
3
), pp.
257
265
15.
Liu
,
H.
, and
Dai
,
J. S.
,
2002
, “
Carton Manipulation Analysis Using Configuration Transformation
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
216
(
5
), pp.
543
555
.10.1243/0954406021525331
16.
Liu
,
H.
, and
Dai
,
J. S.
,
2003
, “
An Approach to Carton-Folding Trajectory Planning Using Dual Robotic Fingers
,”
Rob. Auton. Syst.
,
42
(
1
), pp.
47
63
.10.1016/S0921-8890(02)00312-3
17.
Europe, P. C., 2013, Carton industry basic information, July. Available at http://www.procarton.com/?section=cartons
18.
Nagasawa
,
S.
,
Fukuzawa
,
Y.
,
Yamaguchi
,
T.
,
Tsukatani
,
S.
, and
Katayama
,
I.
,
2003
, “
Effect of Crease Depth and Crease Deviation on Folding Deformation Characteristics of Coated Paperboard
,”
J. Mater. Process. Technol.
,
140
(
1
), pp.
157
162
.10.1016/S0924-0136(03)00825-2
19.
Beex
,
L.
, and
Peerlings
,
R.
,
2009
, “
An Experimental and Computational Study of Laminated Paperboard Creasing and Folding
,”
Int. J. Solids Struct.
,
46
(
24
), pp.
4192
4207
.10.1016/j.ijsolstr.2009.08.012
20.
Nygårds
,
M.
,
Just
,
M.
, and
Tryding
,
J.
,
2009
, “
Experimental and Numerical Studies of Creasing of Paperboard
,”
Int. J. Solids Struct.
,
46
(
11
), pp.
2493
2505
.10.1016/j.ijsolstr.2009.02.014
21.
Hicks
,
B. J.
,
Mullineux
,
G.
, and
Sirkett
,
D.
,
2009
, “
A Finite Element-Based Approach for Whole-System Simulation of Packaging Systems for Their Improved Design and Operation
,”
Packag. Technol. Sci.
,
22
(
4
), pp.
209
227
.10.1002/pts.846
22.
Cannella
,
F.
, and
Dai
,
J. S.
,
2006
, “
Crease Stiffness and Panel Compliance of Carton Folds and Their Integration in Modelling
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
220
(
6
), pp.
847
855
.10.1243/09544062JMES242
23.
Dai
,
J. S.
, and
Cannella
,
F.
,
2008
, “
Stiffness Characteristics of Carton Folds for Packaging
,”
ASME J. Mech. Des.
,
130
(
2
), p. 022305.10.1115/1.2813785
24.
Sirkett
,
D.
,
Hicks
,
B.
,
Berry
,
C.
,
Mullineux
,
G.
, and
Medland
,
A.
,
2006
, “
Simulating the Behaviour of Folded Cartons During Complex Packing Operations
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
220
(
12
), pp.
1797
1811
.10.1243/0954406JMES109
25.
Howell
,
L. L.
,
2001
, Compliant Mechanisms, John Wiley and Sons, New York.
26.
Winder
,
B. G.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2009
, “
Kinematic Representations of Pop-Up Paper Mechanisms
,”
ASME J. Mech. Rob
,
1
(
2
),
p. 021099
.10.1115/1.3046128
27.
Greenberg
,
H.
,
Gong
,
M.
,
Magleby
,
S.
, and
Howell
,
L.
,
2011
, “
Identifying Links Between Origami and Compliant Mechanisms
,”
Mech. Sci.
,
2
, pp. 217–225.10.5194/ms-2-217-2011
28.
Gollnick
,
P. S.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2011
, “
An Introduction to Multilayer Lamina Emergent Mechanisms
,”
ASME J. Mech. Des.
,
133
(8), p.
081006
.10.1115/1.4004542
29.
Ball
,
R. S.
,
1900
,
A Treatise on the Theory of Screws
,
Cambridge University
,
Cambridge, UK
.
30.
Murray
,
R. M.
,
Li
,
Z.
,
Sastry
,
S. S.
, and
Sastry
,
S. S.
,
1994
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
,
Boca Raton, FL
.
31.
Gosselin
,
C.
,
1990
, “
Stiffness Mapping for Parallel Manipulators
,”
IEEE Trans. Rob. Autom.
,
6
(
3
), pp.
377
382
.10.1109/70.56657
32.
Quennouelle
,
C.
, and
Gosselin
,
C. á.
,
2008
, “
Stiffness Matrix of Compliant Parallel Mechanisms
,”
Advances in Robot Kinematics: Analysis and Design
, J. Lenarčič and P. Wenger, eds.,
Springer
,
New York
, pp.
331
341
.
33.
Wolf
,
S.
, and
Hirzinger
,
G.
,
2008
, “
A New Variable Stiffness Design: Matching Requirements of the Next Robot Generation
,”
IEEE International Conference on Robotics and Automation
, ICRA
2008
, IEEE, New York, pp.
1741
1746
.
34.
Ham
,
R. V.
,
Sugar
,
T.
,
Vanderborght
,
B.
,
Hollander
,
K.
, and
Lefeber
,
D.
,
2009
, “
Compliant Actuator Designs
,”
IEEE Rob. Autom. Mag.
,
16
(
3
), pp.
81
94
.10.1109/MRA.2009.933629
35.
Carpino
,
G.
,
Accoto
,
D.
,
Sergi
,
F.
,
Tagliamonte
,
N. L.
, and
Guglielmelli
,
E.
,
2012
, “
A Novel Compact Torsional Spring for Series Elastic Actuators for Assistive Wearable Robots
,”
ASME J. Mech. Des.
,
134
(
12
), p.
121002
.10.1115/1.4007695