Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Large-scale automated investigation of free-falling paper shapes via iterative physical experimentation

Nature Machine Intelligence volume 2, pages 68–75 (2020)Cite this article

Subjects

Abstract

Free-falling paper shapes exhibit rich, complex and varied behaviours that are extremely challenging to model analytically. Physical experimentation aids in system understanding, but is time-consuming, sensitive to initial conditions and reliant on subjective visual behavioural classification. In this study, robotics, computer vision and machine learning are used to autonomously fabricate, drop, analyse and classify the behaviours of hundreds of shapes. The system is validated by reproducing results for falling discs, which exhibit four falling styles: tumbling, chaotic, steady and periodic. A previously determined mapping from a non-dimensional parameter space to behaviour groups is shown to be consistent with these new experiments for tumbling and chaotic behaviours. However, steady or periodic behaviours are observed in previously unseen areas of the parameter space. More complex hexagon, square and cross shapes are investigated, showing that the non-dimensional parameter space generalizes to these shapes. The system highlights the potential of robotics for the investigation of complex physical systems, of which falling paper is one example, and provides a template for future investigation of such systems.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic of the falling-paper system and proposed iterative physical experimentation approach.
Fig. 2: Diagram of the experimental set-up.
Fig. 3: Time-lapse images of the three falling behaviours observed in each shape.
Fig. 4: Trajectories and observable area profiles of falling shapes for different automatically classified behavioural groups.
Fig. 5: Variation in measured falling characteristics with respect to design parameters and automatically classified behaviour groups.
Fig. 6: Automatically classified falling behaviours in Reynolds number \(\mathrm{Re}\) and non-dimensional moment of inertia \({I}^{* }\) parameter space.

Data availability

Example data are available at https://github.com/th533/Falling-Paper.

Code availability

Example code is available at https://github.com/th533/Falling-Paper.

References

  1. Varshney, K., Chang, S. & Wang, Z. J. The kinematics of falling maple seeds and the initial transition to a helical motion. Nonlinearity 25, C1 (2012).

    MATH  Google Scholar 

  2. Norberg, R. A. Autorotation, self stability and structure of single winged fruits and seeds (samaras) with comparative remarks on animal flight. Biol. Rev. 48, 561–596 (1973).

    Google Scholar 

  3. Mikaelian, K. O. Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80, 508–511 (1998).

    Google Scholar 

  4. Epstein, I. R. & Showalter, K. Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem. 100, 13132–13147 (1996).

    Google Scholar 

  5. Nicolis, G Introduction to Nonlinear Science (Cambridge Univ. Press, 1995).

  6. May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).

    MATH  Google Scholar 

  7. Mustapha, H. & Dimitrakopoulos, R. High-order stochastic simulation of complex spatially distributed natural phenomena. Math. Geosci. 42, 457–485 (2010).

    MathSciNet  MATH  Google Scholar 

  8. Brodbeck, L., Hauser, S. & Iida, F. Morphological evolution of physical robots through model-free phenotype development. PLoS One 10, e0128444 (2015).

    Google Scholar 

  9. Vujovic, V., Rosendo, A., Brodbeck, L. & Iida, F. Evolutionary developmental robotics: improving morphology and control of physical robots. Artif. Life 23, 169–185 (2017).

    Google Scholar 

  10. Rieffel, J., Knox, D., Smith, S. & Trimmer, B. Growing and evolving soft robots. Artif. Life 20, 143–162 (2014).

    Google Scholar 

  11. Cheney, N., MacCurdy, R., Clune, J. & Lipson, H. Unshackling evolution. ACM SIGEVOlution 7, 11–23 (2014).

    Google Scholar 

  12. Rosendo, A., vonAtzigen, M. & Iida, F. The trade-off between morphology and control in the co-optimized design of robots. PLoS One 12, e0186107 (2017).

    Google Scholar 

  13. Saar, K. A., Giardina, F. & Iida, F. Model-free design optimization of a hopping robot and its comparison with a human designer. IEEE Robot. Autom. Lett. 3, 1245–1251 (2018).

    Google Scholar 

  14. Nakajima, K., Hauser, H., Li, T. & Pfeifer, R. Information processing via physical soft body. Sci. Rep. 5, 10487 (2015).

    Google Scholar 

  15. Hauser, H., Ijspeert, A. J., Füchslin, R. M., Pfeifer, R. & Maass, W. Towards a theoretical foundation for morphological computation with compliant bodies. Biol. Cybern. 105, 355–370 (2011).

    MathSciNet  MATH  Google Scholar 

  16. Maxwell, J. C. On a particular case of the descent of a heavy body in a resisting medium. Camb. Dublin Math. 9, 145–148 (1854).

    Google Scholar 

  17. Field, S. B., Klaus, M., Moore, M. G. & Nori, F. Chaotic dynamics of falling disks. Nature 388, 252–254 (1997).

    Google Scholar 

  18. Zhong, H., Chen, S. & Lee, C. Experimental study of freely falling thin disks: transition from planar zigzag to spiral. Phys. Fluids 23, 011702 (2011).

    Google Scholar 

  19. Lee, C. et al. Experimental investigation of freely falling thin disks. Part 2. Transition of three-dimensional motion from zigzag to spiral. J. Fluid Mech. 732, 77–104 (2013).

    MATH  Google Scholar 

  20. Heisinger, L., Newton, P. & Kanso, E. Coins falling in water. J. Fluid Mech. 742, 243–253 (2014).

    Google Scholar 

  21. Stringham, G., Simons, D. & Guy, H. The Behavior of Large Particles Falling in Quiescent Liquids (US Government Printing Office, 1969).

  22. Willmarth, W., Hawk, N. & Harvey, R. Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208 (1964).

    MATH  Google Scholar 

  23. Mahadevan, L., Ryu, W. S. & Samuel, A. D. Tumbling cards. Phys. Fluids 11, 1–3 (1999).

    MATH  Google Scholar 

  24. Skews, B. W. Autorotation of rectangular plates. J. Fluid Mech. 217, 33–40 (1990).

    Google Scholar 

  25. Wang, W. B., Hu, R. F., Xu, S. J. & Wu, Z. N. Influence of aspect ratio on tumbling plates. J. Fluid Mech. 733, 650–679 (2013).

    MATH  Google Scholar 

  26. Vincent, L., Shambaugh, W. S. & Kanso, E. Holes stabilize freely falling coins. J. Fluid Mech. 801, 250–259 (2016).

    Google Scholar 

  27. Varshney, K., Chang, S. & Wang, Z. J. Unsteady aerodynamic forces and torques on falling parallelograms in coupled tumbling-helical motions. Phys. Rev. E 87, 053021 (2013).

    Google Scholar 

  28. Belmonte, A., Eisenberg, H. & Moses, E. From flutter to tumble: inertial drag and Froude similarity in falling paper. Phys. Rev. Lett. 81, 345–348 (1998).

    Google Scholar 

  29. Andersen, A., Pesavento, U. & Wang, Z. J. Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91–104 (2005).

    MathSciNet  MATH  Google Scholar 

  30. Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. On the zigzag dynamics of freely moving axisymmetric bodies. Phys. Fluids 17, 098107 (2005).

    MathSciNet  MATH  Google Scholar 

  31. Pesavento, U. & Wang, Z. J. Falling paper: Navier–Stokes solutions, model of fluid forces and center of mass elevation. Phys. Rev. Lett. 93, 144501 (2004).

    Google Scholar 

  32. Jin, C. & Xu, K. Numerical study of the unsteady aerodynamics of freely falling plates. Commun. Comput. Phys. 3, 834–851 (2008).

    MathSciNet  Google Scholar 

  33. Waltz, B. & Buchanan, B. G. Automating science. Science 324, 43–44 (2009).

    Google Scholar 

  34. Peplow, M. Organic synthesis: the robot-chemist. Nature 512, 20–22 (2014).

    Google Scholar 

  35. Mjolsness, E. & DeCoste, D. Machine learning for science: state of the art and future prospects. Science 293, 2051–2055 (2001).

    Google Scholar 

  36. Bottou, L., Curtis, F. E. & Nocedal, J. Optimization methods for large-scale machine learning. SIAM Rev. 60, 223–231 (2018).

    MathSciNet  MATH  Google Scholar 

  37. Soldatova, L. N., Clare, A., Sparkes, A. & King, R. D. An ontology for a robot scientist. Bioinformatics 22, e464–e471 (2006).

    Google Scholar 

  38. Sparkes, A. et al. Towards robot scientists for autonomous scientific discovery. Automated Exp. 2, 1 (2010).

    Google Scholar 

  39. Fan, D. et al. A robotic intelligent towing tank for learning complex fluid-structure dynamics. Sci. Robot. 4, eaay5063 (2019).

    Google Scholar 

  40. Chapman, T. Lab automation and robotics: automation on the move. Nature 421, 661–663 (2003).

    Google Scholar 

  41. Kachel, V., Sindelar, G. & Grimm, S. High-throughput isolation of ultra-pure plasmid DNA by a robotic system. BMC Biotechnol. 6, 9 (2006).

    Google Scholar 

  42. Sparkes, A. et al. An integrated laboratory robotic system for autonomous discovery of gene function. J. Assoc. Lab. Automat. 15, 33–40 (2010).

    Google Scholar 

  43. King, R. D. et al. Functional genomic hypothesis generation and experimentation by a robot scientist. Nature 427, 247–252 (2004).

    Google Scholar 

  44. Vasilevich, A. & de Boer, J. Robot-scientists will lead tomorrowas biomaterials discovery. Curr. Opin. Biomed. Eng. 6, 74–80 (2018).

    Google Scholar 

  45. Bellemare, M. et al. Unifying count-based exploration and intrinsic motivation. In Proceedings of Neural Information Processing Systems 29 1471–1479 (NIPS, 2016).

  46. Tang, H. et al. Exploration: a study of count-based exploration for deep reinforcement learning. In Proceedings of Neural Information Processing Systems 30 2753–2762 (NIPS, 2017).

  47. Frankel, F. & Reid, R. Big data: distilling meaning from data. Nature 455, 30 (2008).

    Google Scholar 

  48. Howison, T., Hughes, J., Giardina, F. & Iida, F. Physics driven behavioural clustering of free-falling paper shapes. PLoS One 14, e0217997 (2019).

    Google Scholar 

  49. Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 309, 1236–1239 (2005).

    MathSciNet  Google Scholar 

  50. Bongard, J. & Lipson, H. Automated reverse engineering of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 104, 9943–9948 (2007).

    MATH  Google Scholar 

  51. Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113, 3932–3937 (2016).

    MathSciNet  MATH  Google Scholar 

  52. Hartigan, J. A. & Wong, M. A. Algorithm AS 136: a k-means clustering algorithm. Appl. Stat. 28, 100–108 (1979).

    MATH  Google Scholar 

  53. Goldberg, D. E. & Holland, J. H. Genetic algorithms and machine learning. Mach. Learn. 3, 95–99 (1988).

    Google Scholar 

  54. Mouret, J. B. & Clune, J. Illuminating search spaces by mapping elites. Preprint at https://arxiv.org/abs/1504.04909 (2015).

Download references

Acknowledgements

We acknowledge funding from EPSRC RG92738 and The Mathworks Ltd.

Author information

Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Cambridge, UK

    Toby Howison, Josie Hughes & Fumiya Iida

Authors
  1. Toby Howison
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josie Hughes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fumiya Iida
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

T.H., J.H. and F.I. conceived the study. T.H. designed the experimental set-up. T.H. and J.H. managed the experiments. T.H. and J.H. analysed the data. T.H., J.H. and F.I. wrote the manuscript.

Corresponding author

Correspondence to Toby Howison.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Material

Supplementary material containing Supplementary Figs. 1–7 and Tables 1–4.

Supplementary Video 1

Demonstration of iterative physical experimentation system showing manufacture, picking, dropping and analysis of falling paper shapes.

Supplementary Video 2

Slow-motion representative examples of steady and periodic, chaotic and tumbling behaviours of circles, hexagons, squares and crosses.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Howison, T., Hughes, J. & Iida, F. Large-scale automated investigation of free-falling paper shapes via iterative physical experimentation. Nat Mach Intell 2, 68–75 (2020). https://doi.org/10.1038/s42256-019-0135-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42256-019-0135-z

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing