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 Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$119.00 per year
only $9.92 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Example data are available at https://github.com/th533/Falling-Paper.
Example code is available at https://github.com/th533/Falling-Paper.
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).
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).
Mikaelian, K. O. Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80, 508–511 (1998).
Epstein, I. R. & Showalter, K. Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem. 100, 13132–13147 (1996).
Nicolis, G Introduction to Nonlinear Science (Cambridge Univ. Press, 1995).
May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).
Mustapha, H. & Dimitrakopoulos, R. High-order stochastic simulation of complex spatially distributed natural phenomena. Math. Geosci. 42, 457–485 (2010).
Brodbeck, L., Hauser, S. & Iida, F. Morphological evolution of physical robots through model-free phenotype development. PLoS One 10, e0128444 (2015).
Vujovic, V., Rosendo, A., Brodbeck, L. & Iida, F. Evolutionary developmental robotics: improving morphology and control of physical robots. Artif. Life 23, 169–185 (2017).
Rieffel, J., Knox, D., Smith, S. & Trimmer, B. Growing and evolving soft robots. Artif. Life 20, 143–162 (2014).
Cheney, N., MacCurdy, R., Clune, J. & Lipson, H. Unshackling evolution. ACM SIGEVOlution 7, 11–23 (2014).
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).
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).
Nakajima, K., Hauser, H., Li, T. & Pfeifer, R. Information processing via physical soft body. Sci. Rep. 5, 10487 (2015).
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).
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).
Field, S. B., Klaus, M., Moore, M. G. & Nori, F. Chaotic dynamics of falling disks. Nature 388, 252–254 (1997).
Zhong, H., Chen, S. & Lee, C. Experimental study of freely falling thin disks: transition from planar zigzag to spiral. Phys. Fluids 23, 011702 (2011).
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).
Heisinger, L., Newton, P. & Kanso, E. Coins falling in water. J. Fluid Mech. 742, 243–253 (2014).
Stringham, G., Simons, D. & Guy, H. The Behavior of Large Particles Falling in Quiescent Liquids (US Government Printing Office, 1969).
Willmarth, W., Hawk, N. & Harvey, R. Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208 (1964).
Mahadevan, L., Ryu, W. S. & Samuel, A. D. Tumbling cards. Phys. Fluids 11, 1–3 (1999).
Skews, B. W. Autorotation of rectangular plates. J. Fluid Mech. 217, 33–40 (1990).
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).
Vincent, L., Shambaugh, W. S. & Kanso, E. Holes stabilize freely falling coins. J. Fluid Mech. 801, 250–259 (2016).
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).
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).
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).
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. On the zigzag dynamics of freely moving axisymmetric bodies. Phys. Fluids 17, 098107 (2005).
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).
Jin, C. & Xu, K. Numerical study of the unsteady aerodynamics of freely falling plates. Commun. Comput. Phys. 3, 834–851 (2008).
Waltz, B. & Buchanan, B. G. Automating science. Science 324, 43–44 (2009).
Peplow, M. Organic synthesis: the robot-chemist. Nature 512, 20–22 (2014).
Mjolsness, E. & DeCoste, D. Machine learning for science: state of the art and future prospects. Science 293, 2051–2055 (2001).
Bottou, L., Curtis, F. E. & Nocedal, J. Optimization methods for large-scale machine learning. SIAM Rev. 60, 223–231 (2018).
Soldatova, L. N., Clare, A., Sparkes, A. & King, R. D. An ontology for a robot scientist. Bioinformatics 22, e464–e471 (2006).
Sparkes, A. et al. Towards robot scientists for autonomous scientific discovery. Automated Exp. 2, 1 (2010).
Fan, D. et al. A robotic intelligent towing tank for learning complex fluid-structure dynamics. Sci. Robot. 4, eaay5063 (2019).
Chapman, T. Lab automation and robotics: automation on the move. Nature 421, 661–663 (2003).
Kachel, V., Sindelar, G. & Grimm, S. High-throughput isolation of ultra-pure plasmid DNA by a robotic system. BMC Biotechnol. 6, 9 (2006).
Sparkes, A. et al. An integrated laboratory robotic system for autonomous discovery of gene function. J. Assoc. Lab. Automat. 15, 33–40 (2010).
King, R. D. et al. Functional genomic hypothesis generation and experimentation by a robot scientist. Nature 427, 247–252 (2004).
Vasilevich, A. & de Boer, J. Robot-scientists will lead tomorrowas biomaterials discovery. Curr. Opin. Biomed. Eng. 6, 74–80 (2018).
Bellemare, M. et al. Unifying count-based exploration and intrinsic motivation. In Proceedings of Neural Information Processing Systems 29 1471–1479 (NIPS, 2016).
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).
Frankel, F. & Reid, R. Big data: distilling meaning from data. Nature 455, 30 (2008).
Howison, T., Hughes, J., Giardina, F. & Iida, F. Physics driven behavioural clustering of free-falling paper shapes. PLoS One 14, e0217997 (2019).
Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 309, 1236–1239 (2005).
Bongard, J. & Lipson, H. Automated reverse engineering of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 104, 9943–9948 (2007).
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).
Hartigan, J. A. & Wong, M. A. Algorithm AS 136: a k-means clustering algorithm. Appl. Stat. 28, 100–108 (1979).
Goldberg, D. E. & Holland, J. H. Genetic algorithms and machine learning. Mach. Learn. 3, 95–99 (1988).
Mouret, J. B. & Clune, J. Illuminating search spaces by mapping elites. Preprint at https://arxiv.org/abs/1504.04909 (2015).
We acknowledge funding from EPSRC RG92738 and The Mathworks Ltd.
The authors declare no competing interests.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary material containing Supplementary Figs. 1–7 and Tables 1–4.
Demonstration of iterative physical experimentation system showing manufacture, picking, dropping and analysis of falling paper shapes.
Slow-motion representative examples of steady and periodic, chaotic and tumbling behaviours of circles, hexagons, squares and crosses.
About this article
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