0
Research Papers

# Kinematic Analysis and Stiffness Validation of Origami Cartons

[+] Author and Article Information
C. Qiu

e-mail: chen.qiu@kcl.ac.uk

Postdoctoral Research Associate
Centre for Robotic Research,
Kings's College London,
London WC2R 2LS, UK

Jian S. Dai

Chair of Mechanisms and Robotics,
MoE Key Laboratory for Mechanism Theory and Equipment Design,
Tianjin University,
Tianjin, China
Centre for Robotic Research,
Kings's College London,
London WC2R 2LS, UK
e-mail: jian.dai@kcl.ac.uk

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 7, 2013; final manuscript received August 28, 2013; published online September 27, 2013. Assoc. Editor: Larry L. Howell.

J. Mech. Des 135(11), 111004 (Sep 27, 2013) (13 pages) Paper No: MD-13-1072; doi: 10.1115/1.4025381 History: Received February 07, 2013; Revised August 28, 2013

## Abstract

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

Kanade, T., 1980, “A Theory of Origami World,” Artif. Intell., 13(3), pp. 279–311.
Lang, R. J., and Hull, T. C., 2005, “Origami Design Secrets: Mathematical Methods for an Ancient Art,” Math. Intell., 27(2), pp. 92–95.
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.
Dai, J. S., and Jones, J., 1999, “Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds,” ASME J. Mech. Des., 121(3), pp. 375–382.
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.
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.
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.
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.
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.
Lu, L., and Akella, S., 2000, “Folding Cartons With Fixtures: A Motion Planning Approach,” IEEE Trans. Rob. Autom., 16(4), pp. 346–356.
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.
Balkcom, D., and Mason, M., 2008, “Robotic Origami Folding,” Int. J. Robot. Res., 27(5), pp. 613–627.
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.
Mullineux, G., and Matthews, J., 2010, “Constraint-Based Simulation of Carton Folding Operations,” Comput.-Aided Des., 42(3), pp. 257–265.
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.
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.
Europe, P. C., 2013, Carton industry basic information, July. Available at http://www.procarton.com/?section=cartons
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.
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.
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.
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.
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.
Dai, J. S., and Cannella, F., 2008, “Stiffness Characteristics of Carton Folds for Packaging,” ASME J. Mech. Des., 130(2), p. 022305.
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.
Howell, L. L., 2001, Compliant Mechanisms, John Wiley and Sons, New York.
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.
Greenberg, H., Gong, M., Magleby, S., and Howell, L., 2011, “Identifying Links Between Origami and Compliant Mechanisms,” Mech. Sci., 2, pp. 217–225.
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.
Ball, R. S., 1900, A Treatise on the Theory of Screws, Cambridge University, Cambridge, UK.
Murray, R. M., Li, Z., Sastry, S. S., and Sastry, S. S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
Gosselin, C., 1990, “Stiffness Mapping for Parallel Manipulators,” IEEE Trans. Rob. Autom., 6(3), pp. 377–382.
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.
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.
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.
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.

## Figures

Fig. 4

Tray carton mechanism and its geometric relationship. (a) Tray carton mechanism and (b) geometry of a tray carton.

Fig. 3

Carton corner linkage models in three origami-type cartons

Fig. 2

Folding stages of different origami-type cartons

Fig. 1

Origami-type cartons and tested carton samples. (a) A group of origami-type cartons, tray carton in the left, crash-lock carton in the middle, and gable carton in the right and (b) origami carton samples, and they correspond to the origami cartons used in food packaging industry.

Fig. 8

Folding moment and stiffness comparisons of virgin and repeated folding process. (a) Folding moment comparison between virgin and repeated folding, solid and dashed lines represent experiment results, and dashed-dotted lines represent ±1 standard deviation. (b) Folding stiffness comparison, solid and dashed lines represent experiment results, error bars represent standard deviations within ±5 deg range of eight folding positions every 10 deg from 10 deg to 80 deg.

Fig. 10

Folding moment and stiffness comparisons with different folding velocities. (a) Folding moment comparison, dashed line, dashed-dotted line, and solid line correspond to experiment results, and dotted lines represent ±1 standard deviation. (b) Folding stiffness comparison, dashed line, dashed-dotted line, and solid line correspond to experiment results, seven folding positions every 10 deg from 20 deg to 80 deg are selected to represent motion-stiffness curves.

Fig. 11

Folding moment and stiffness comparisons of tray carton. (a) Folding moment comparison, dashed line represents mathematical model, solid line represents experiment result. (b) Folding stiffness comparison, dashed line represents mathematical model, solid line represents experiment result.

Fig. 5

Gable carton mechanism and geometric relationship. (a) Gable carton mechanism and (b) geometry of a gable carton.

Fig. 6

Crash-lock carton mechanism and geometric relationship. (a) Crash-lock carton mechanism and (b) geometry of a crash-lock carton.

Fig. 7

Experiment setup

Fig. 9

Folding moment and stiffness comparisons with different crease lengths. (a) Folding moment comparison, dashed line, dashed-dotted line and solid line corresponds to experiment results, and dotted lines represent ±1 standard deviation. (b) Folding stiffness comparison, dashed line, dashed-dotted line and solid line correspond to experiment results, seven folding positions every 10 deg from 20 deg to 80 deg are selected to represent motion-stiffness curves.

Fig. 13

Folding moment and stiffness comparisons of crash-lock carton. (a) Folding moment comparison, dashed line represents mathematical model, solid line represents experiment result. (b) Folding stiffness comparison, dashed line represents mathematical model, solid line represents experiment result.

Fig. 12

Folding moment and stiffness comparisons of gable-style carton. (a) Folding moment comparison, dashed line represents mathematical model, solid line represents experiment result. (b) Folding stiffness comparison, dashed line represents mathematical model, solid line represents experiment result.

Fig. 14

Carton equivalent mechanism and the carton corner linkages. (a) Manipulated carton and equivalent mechanism and (b) corner linkages A.

Fig. 15

Folding moment and stiffness comparisons of single crease with various crease lengths. (a) Folding moment comparison, dashed line, dashed-dotted line, and solid line correspond to experiment results, and dotted lines represent ±1 standard deviation. (b) Folding stiffness comparison, dashed line, dashed-dotted line, and solid line correspond to experiment results, 12 folding positions every 6 deg from 10 deg to 76 deg are selected to represent motion-stiffness curves.

Fig. 16

Folding moment and stiffness comparisons between mathematical model and experimental data. (a) Folding moment comparison, dashed line represents mathematical model, solid line represents experiment result. (b) Folding stiffness comparison, dashed line represents mathematical model, solid line represents experiment result.

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections