Abstract
Characterizing forces on deformable objects intruding into sand and soil requires understanding the solid- and fluid-like responses of such substrates and their effect on the state of the object. The most detailed studies of intrusion in dry granular media have revealed that interactions of fixed-shape objects during free impact (for example, cannonballs) and forced slow penetration can be described by hydrostatic- and hydrodynamic-like forces. Here we investigate a new class of granular interactions: rapid intrusions by objects that change shape (self-deform) through passive and active means. Systematic studies of a simple spring-mass robot jumping on dry granular media reveal that jumping performance is explained by an interplay of nonlinear frictional and hydrodynamic drag as well as induced added mass (unaccounted by traditional intrusion models) characterized by a rapidly solidified region of grains accelerated by the foot. A model incorporating these dynamics reveals that added mass degrades the performance of certain self-deformations owing to a shift in optimal timing during push-off. Our systematic robophysical experiment reveals both new soft-matter physics and principles for robotic self-deformation and control, which together provide principles of movement in deformable terrestrial environments.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Alexander, R. M. Principles of Animal Locomotion (Princeton Univ. Press, 2003).
Blickhan, R. The spring-mass model for running and hopping. J. Biomech. 22, 1217–1227 (1989).
Raibert, M. Legged Robots that Balance (MIT Press, 1986).
Pratt, G. A. & Williamson, M. M. Intelligent Robots and Systems 95. ’Human Robot Interaction and Cooperative Robots’, Proceedings. 1995 IEEE/RSJ International Conference on Vol. 1, 399–406 (IEEE, 1995).
Komsuoglu, H., Majumdar, A., Aydin, Y. O. & Koditschek, D. E. in Experimental Robotics (eds Khatib, O., Kumar, V. & Sukhatme, G.) 667–684 (Springer, 2014).
Qian, F. et al. Walking and Running on Yielding and Fluidizing Ground 345–353 (RSS, 2013); http://www.roboticsproceedings.org/rss08/p44.html.
Bridge, B., Dubowsky, S., Kesner, S., Plante, J.-S. & Boston, P. Hopping mobility concept for search and rescue robots. Ind. Robot Int. J. 35, 238–245 (2008).
Burdick, J. & Fiorini, P. Minimalist jumping robots for celestial exploration. Int. J. Robot. Res. 22, 653–674 (2003).
Qian, F. et al. Principles of appendage design in robots and animals determining terradynamic performance on flowable ground. Bioinspir. Biomim. 10, 056014 (2015).
Li, C., Zhang, T. & Goldman, D. I. A terradynamics of legged locomotion on granular media. Science 339, 1408–1412 (2013).
Zhang, T. et al. Ground fluidization promotes rapid running of a lightweight robot. Int. J. Robot. Res. 32, 859–869 (2013).
Li, C., Hsieh, S. T. & Goldman, D. I. Multi-functional foot use during running in the zebra-tailed lizard (callisaurus draconoides). J. Exp. Biol. 215, 3293–3308 (2012).
Moritz, C. T. & Farley, C. T. Human hopping on very soft elastic surfaces: Implications for muscle pre-stretch and elastic energy storage in locomotion. J. Exp. Biol. 208, 939–949 (2005).
Maladen, R. D., Ding, Y., Umbanhowar, P. B., Kamor, A. & Goldman, D. I. Mechanical models of sandfish locomotion reveal principles of high performance subsurface sand-swimming. J. R. Soc. Interface 8, 1332–1345 (2011).
Maladen, R. D., Ding, Y., Li, C. & Goldman, D. I. Undulatory swimming in sand: Subsurface locomotion of the sandfish lizard. Science 325, 314–318 (2009).
Katsuragi, H. & Durian, D. J. Unified force law for granular impact cratering. Nature Phys. 3, 420–423 (2007).
Tsimring, L. & Volfson, D. Modeling of impact cratering in granular media. Powders Grains 2, 1215–1223 (2005).
Umbanhowar, P. & Goldman, D. Granular impact and the critical packing state. Phys. Rev. E 82, 010301(R) (2010).
Euler, L. Neue Grundsätze der Artillerie; reprinted in Euler’s Opera Omnia Vol. 2, 1922 (Druck und Verlag Von B.G. Teubner, 1745).
Poncelet, J. V. Cours de Mécanique Industrielle (Lithographie de Clouet, Paris, 1829).
Robins, B. & Curtis, W. New Principles of Gunnery (Richmond Publishing Company Limited, 1972).
Backman, M. E. & Goldsmith, W. The mechanics of penetration of projectiles into targets. Int. J. Eng. Sci. 16, 1–99 (1978).
Allen, W. A., Mayfield, E. B. & Morrison, H. L. Dynamics of a projectile penetrating sand. J. Appl. Phys. 28, 370–376 (1957).
Forrestal, M. & Luk, V. Penetration into soil targets. Int. J. Impact Eng. 12, 427–444 (1992).
Pouliquen, O. & Forterre, Y. A non-local rheology for dense granular flows. Phil. Trans. R. Soc. A 367, 5091–5107 (2009).
Waitukaitis, S. R. & Jaeger, H. M. Impact-activated solidification of dense suspensions via dynamic jamming fronts. Nature 487, 205–209 (2012).
Katsuragi, H. & Durian, D. J. Drag force scaling for penetration into granular media. Phys. Rev. E 87, 052208 (2013).
Brennen, C. A Review of Added Mass and Fluid Inertial Forces Tech. Rep. (Defense Technical Information Center (DTIC), 1982).
Pandy, M. G., Zajac, F. E., Sim, E. & Levine, W. S. An optimal control model for maximum-height human jumping. J. Biomech. 23, 1185–1198 (1990).
Zajac, F. E. Muscle coordination of movement: A perspective. J. Biomech. 26, 109–124 (1993).
Aguilar, J., Lesov, A., Wiesenfeld, K. & Goldman, D. I. Lift-off dynamics in a simple jumping robot. Phys. Rev. Lett. 109, 174301 (2012).
Gravish, N., Umbanhowar, P. B. & Goldman, D. I. Force and flow at the onset of drag in plowed granular media. Phys. Rev. E 89, 042202 (2014).
Tapia, F., Espíndola, D., Hamm, E. & Melo, F. Effect of packing fraction on shear band formation in a granular material forced by a penetrometer. Phys. Rev. E 87, 014201 (2013).
Stone, M. B. et al. Stress propagation: Getting to the bottom of a granular medium. Nature 427, 503–504 (2004).
Stone, M. et al. Local jamming via penetration of a granular medium. Phys. Rev. E 70, 041301 (2004).
Le Bouil, A., Amon, A., McNamara, S. & Crassous, J. Emergence of cooperativity in plasticity of soft glassy materials. Phys. Rev. Lett. 112, 246001 (2014).
Glasheen, J. & McMahon, T. A hydrodynamic model of locomotion in the basilisk lizard. Nature 380, 340–341 (1996).
Richardson, E. The impact of a solid on a liquid surface. Proc. Phys. Soc. 61, 352–367 (1948).
Wagner, H. Phenomena associated with impacts and sliding on liquid surfaces. Z. Angew. Math. Mech. 12, 193–215 (1932).
Sakakibara, J., Nakagawa, M. & Yoshida, M. Stereo-PIV study of flow around a maneuvering fish. Exp. Fluids 36, 282–293 (2004).
Clark, A. H. & Behringer, R. P. Granular impact model as an energy-depth relation. Europhys. Lett. 101, 64001 (2013).
Zhang, J., Johansson, K. H., Lygeros, J. & Sastry, S. Zeno hybrid systems. Int. J. Robust Nonlin. Control 11, 435–451 (2001).
Acknowledgements
This work was supported by NSF Physics of Living Systems, Burroughs Wellcome Fund, and the Army Research Office. We thank A. Karsai for assistance in simulation work and P. Umbanhowar and L. London for insightful comments and discussion.
Author information
Authors and Affiliations
Contributions
J.A. and D.I.G. conceived the study and wrote the paper. J.A. performed the experimental work, designed and ran the simulation models, and analysed the results.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Movie 1
Supplementary Movie (AVI 13521 kb)
Supplementary Movie 2
Supplementary Movie (AVI 1230 kb)
Rights and permissions
About this article
Cite this article
Aguilar, J., Goldman, D. Robophysical study of jumping dynamics on granular media. Nature Phys 12, 278–283 (2016). https://doi.org/10.1038/nphys3568
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys3568
This article is cited by
-
Swift progress for robots over complex terrain
Nature (2023)
-
Measuring the mechanical properties of small body regolith layers using a granular penetrometer
Astrodynamics (2023)
-
Constant speed penetration into granular materials: drag forces from the quasistatic to inertial regime
Granular Matter (2021)
-
Stagnant zone formation in a 2D bed of circular and elongated grains under penetration
Granular Matter (2020)
-
Fluctuation and self-diffusion research about dry granular materials under shearing
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019)