Undulation enables gliding in flying snakes

Abstract

When flying snakes glide, they use aerial undulation. To determine if aerial undulation is a flight control strategy or a non-functional behavioural vestige of lateral undulation, we measured snake glides using high-speed motion capture and developed a new dynamical model of gliding. Reconstructions of the snake’s wing-body reveal that aerial undulation is composed of horizontal and vertical waves, whose phases differ by 90° and whose frequencies differ by a factor of two. Using these results, we developed a three-dimensional mathematical model of snake flight that incorporates aerodynamic and inertial effects. Although simulated glides without undulation attained some horizontal distance, they are biologically unrealistic because they failed due to roll and pitch instabilities. In contrast, the inclusion of undulation stabilized the rotational motion and markedly increased glide performance. This work demonstrates that aerial undulation in snakes serves a different function than known uses of undulation in other animals, and suggests a new template of control for dynamic flying robots.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Measurements of gliding in flying snakes.
Fig. 2: Measured body kinematics in flying snakes.
Fig. 3: Comparison of measured and simulated snake glides.
Fig. 4: Simulated flying snake glides.
Fig. 5: Effect of dorsoventral bending (dψ) on glide dynamics and control of pitching motion.
Fig. 6: Relative contributions of inertial and aerodynamic moments in the flying snake model.

Data availability

All data that support the plots within this paper and other findings of this study are available in the Data subdirectories of the code repository and at https://drive.google.com/drive/folders/1FpSBUD1XY3guuWjGUE5V7dqluNkoXyKy for processed experiment and simulation data.

Code availability

Code used to analyse the glide trials and perform the glide simulations is available at github.com/TheSochaLab/Undulation-enables-gliding-in-flying-snakes.

References

  1. 1.

    Walton, M., Jayne, B. C. & Bennet, A. F. The energetic cost of limbless locomotion. Science 249, 524–527 (1990).

    ADS  Article  Google Scholar 

  2. 2.

    Hopkins, J. K., Spranklin, B. W. & Gupta, S. K. A survey of snake-inspired robot designs. Bioinspir. Biomim. 4, 021001 (2009).

    ADS  Article  Google Scholar 

  3. 3.

    Guo, Z. V. & Mahadevan, L. Limbless undulatory propulsion on land. Proc. Natl Acad. Sci. USA 105, 3179–3184 (2008).

    ADS  Article  Google Scholar 

  4. 4.

    Hu, D. L., Nirody, J., Scott, T. & Shelley, M. J. The mechanics of slithering locomotion. Proc. Natl Acad. Sci. USA 106, 10081–10085 (2009).

    ADS  Article  Google Scholar 

  5. 5.

    Alben, S. Optimizing snake locomotion in the plane. Proc. R. Soc. A 469, 20130236 (2013).

    ADS  MATH  Article  Google Scholar 

  6. 6.

    Hirose, S. Biologically Inspired Robots: Snake-like Locomotors and Manipulators (Oxford University Press, 1993).

  7. 7.

    Marvi, H. et al. Sidewinding with minimal slip: snake and robot ascent of sandy slopes. Science 346, 224–229 (2014).

    ADS  Article  Google Scholar 

  8. 8.

    Astley, H. C. et al. Modulation of orthogonal body waves enables high maneuverability in sidewinding locomotion. Proc. Natl Acad. Sci. USA 112, 6200–6205 (2015).

    ADS  Article  Google Scholar 

  9. 9.

    Pierce-Shimomura, J. T. et al. Genetic analysis of crawling and swimming locomotory patterns in C. elegans. Proc. Natl Acad. Sci. USA 105, 20982–20987 (2008).

    ADS  Article  Google Scholar 

  10. 10.

    Fang-Yen, C. et al. Biomechanical analysis of gait adaptation in the nematode Caenorhabditis elegans. Proc. Natl Acad. Sci. USA 107, 20323–20328 (2010).

    ADS  Article  Google Scholar 

  11. 11.

    Hums, I. et al. Regulation of two motor patterns enables the gradual adjustment of locomotion strategy in Caenorhabditis elegans. eLife 5, e14116 (2016).

    Article  Google Scholar 

  12. 12.

    Taylor, S. G. Analysis of the swimming of long and narrow animals. Proc. R. Soc. A 214, 158–183 (1952).

    ADS  MATH  Google Scholar 

  13. 13.

    Gazzola, M., Argentina, M. & Mahadevan, L. Gait and speed selection in slender inertial swimmers. Proc. Natl Acad. Sci. USA 112, 3874–3879 (2015).

    ADS  Article  Google Scholar 

  14. 14.

    Ijspeert, A. J. Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21, 642–653 (2008).

    Article  Google Scholar 

  15. 15.

    Ijspeert, A. J., Crespi, A., Ryczko, D. & Cabelguen, J.-M. From swimming to walking with a salamander robot driven by a spinal cord model. Science 315, 1416–1420 (2007).

    ADS  Article  Google Scholar 

  16. 16.

    Socha, J. J., Jafari, F., Munk, Y. & Byrnes, G. How animals glide: from trajectory to morphology. Can. J. Zool. 93, 901–924 (2015).

    Article  Google Scholar 

  17. 17.

    Socha, J. J. Gliding flight in the paradise tree snake. Nature 418, 603–604 (2002).

    ADS  Article  Google Scholar 

  18. 18.

    Socha, J. J., O’Dempsey, T. & LaBarbera, M. A 3-D kinematic analysis of gliding in a flying snake, Chrysopelea paradisi. J. Exp. Biol. 208, 1817–1833 (2005).

    Article  Google Scholar 

  19. 19.

    Socha, J. J. & LaBarbera, M. Effects of size and behavior on aerial performance of two species of flying snakes (Chrysopelea). J. Exp. Biol. 208, 1835–1847 (2005).

    Article  Google Scholar 

  20. 20.

    Socha, J. J., Miklasz, K., Jafari, F. & Vlachos, P. P. Non-equilibrium trajectory dynamics and the kinematics of gliding in a flying snake. Bioinspir. Biomim. 5, 045002 (2010).

    ADS  Article  Google Scholar 

  21. 21.

    Socha, J. J. Gliding flight in Chrysopelea: turning a snake into a wing. Integr. Comp. Biol. 51, 969–982 (2011).

    Article  Google Scholar 

  22. 22.

    Socha, J. J. Becoming airborne without legs: the kinematics of take-off in a flying snake, Chrysopelea paradisi. J. Exp. Biol. 209, 3358–3369 (2006).

    Article  Google Scholar 

  23. 23.

    Holden, D., Socha, J. J., Cardwell, N. D. & Vlachos, P. P. Aerodynamics of the flying snake Chrysopelea paradisi: how a bluff body cross-sectional shape contributes to gliding performance. J. Exp. Biol. 217, 382–394 (2014).

    Article  Google Scholar 

  24. 24.

    Krishnan, A., Socha, J. J., Vlachos, P. P. & Barba, L. A. Lift and wakes of flying snakes. Phys. Fluids 26, 031901 (2014).

    ADS  Article  Google Scholar 

  25. 25.

    Miklasz, K., LaBarbera, M., Chen, X. & Socha, J. J. Effects of body cross-sectional shape on flying snake aerodynamics. Exp. Mech. 50, 1335–1348 (2010).

    Article  Google Scholar 

  26. 26.

    Lillywhite, H. B. How Snakes Work: Structure, Function and Behavior of the World’s Snakes (Oxford University Press, 2014).

  27. 27.

    Gemmell, B. J., Colin, S. P., Costello, J. H. & Dabiri, J. O. Suction-based propulsion as a basis for efficient animal swimming. Nat. Commun. 6, 8790 (2015).

    ADS  Article  Google Scholar 

  28. 28.

    Dudley, R. Mechanisms and implications of animal flight maneuverability. Integr. Comp. Biol. 42, 135–140 (2002).

    Article  Google Scholar 

  29. 29.

    Jafari, F., Tahmasian, S., Ross, S. D. & Socha, J. J. Control of gliding in a flying snake-inspired n-chain model. Bioinspir. Biomim. 12, 066002 (2017).

    ADS  Article  Google Scholar 

  30. 30.

    Feeny, B. F. & Feeny, A. K. Complex modal analysis of the swimming motion of a whiting. J. Vib. Acoust. 135, 021004 (2013).

    Article  Google Scholar 

  31. 31.

    Feeny, B. F., Sternberg, P. W., Cronin, C. J. & Coppola, C. A. Complex orthogonal decomposition applied to nematode posturing. J. Comput. Nonlinear Dyn. 8, 041010 (2013).

    Article  Google Scholar 

  32. 32.

    Ellington, C. P. The aerodynamics of hovering insect flight. I. The quasi-steady analysis. Philos. Trans. R. Soc. B 305, 1–15 (1984).

    ADS  Google Scholar 

  33. 33.

    Dudley, R. & DeVries, P. Tropical rain forest structure and the geographical distribution of gliding vertebrates. Biotropica 22, 432–434 (1990).

    Article  Google Scholar 

  34. 34.

    Moon, B. R. Testing an inference of function from structure: snake vertebrae do the twist. J. Morphol. 241, 217–225 (1999).

    Article  Google Scholar 

  35. 35.

    Saito, M., Fukaya, M. & Iwasaki, T. Modeling, analysis, and synthesis of serpentine locomotion with a multilink robotic snake. IEEE Control Syst. Mag. 22, 64–81 (2002).

    Google Scholar 

  36. 36.

    Full, R. J. & Koditschek, D. E. Templates and anchors: neuromechanical hypotheses of legged locomotion on land. J. Exp. Biol. 202, 3325–3332 (1999).

    Google Scholar 

  37. 37.

    Astley, H. C., Astley, V. E., Brothers, D. & Mendelson, J. R. III Digital analysis of photographs for snake length measurement. Herpetol. Rev. 48, 39–43 (2017).

    Google Scholar 

  38. 38.

    Labbe, R. FilterPy—Kalman filters and other optimal and non-optimal estimation filters in Python (2017); https://github.com/rlabbe/filterpy.

  39. 39.

    Rauch, H. E., Tung, F. & Striebel, C. T. Maximum likelihood estimates of linear dynamic systems. AIAA J. 3, 1445–1450 (1965).

    ADS  MathSciNet  Article  Google Scholar 

  40. 40.

    Winter, D. A. Biomechanics and Motor Control of Human Movement (Wiley, 2009).

  41. 41.

    Knott, G. D. Interpolating Cubic Splines (Birkhäuser, 2000).

  42. 42.

    Jayne, B. C. Muscular mechanisms of snake locomotion: an electromyographic study of lateral undulation of the Florida banded water snake (Nerodia fasciata) and the yellow rat snake (Elaphe obsoleta). J. Morphol. 197, 159–181 (1988).

    Article  Google Scholar 

  43. 43.

    Gillis, G. B. Neuromuscular control of anguilliform locomotion: patterns of red and white muscle activity during swimming in the American eel Anguilla rostrata. J. Exp. Biol. 201, 3245–3256 (1998).

    Google Scholar 

  44. 44.

    Jafari, F., Ross, S. D., Vlachos, P. P. & Socha, J. J. A theoretical analysis of pitch stability during gliding in flying snakes. Bioinspir. Biomim. 9, 025014 (2014).

    ADS  Article  Google Scholar 

  45. 45.

    Yeaton, I. J., Socha, J. J. & Ross, S. D. Global dynamics of non-equilibrium gliding in animals. Bioinspir. Biomim. 12, 026013 (2017).

    ADS  Article  Google Scholar 

  46. 46.

    Haque, A. & Dickman, J. D. Vestibular gaze stabilization: different behavioral strategies for arboreal and terrestrial avians. J. Neurophysiol. 93, 1165–1173 (2005).

    Article  Google Scholar 

  47. 47.

    Yamada, H., Mori, M. & Hirose, S. Stabilization of the head of an undulating snake-like robot. In 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, 3566–3571 (IEEE, 2007).

Download references

Acknowledgements

We thank the Virginia Tech Institute for Creativity, Arts, and Technology (ICAT) for facility access, crew support and funding. In particular, we thank T. Upthegrove, N. McGowan and B. Knapp. We thank G. Nave, T. Weiss, M. Graham, J. Whitehead and J. Garrett for assisting with the kinematics experiments. We thank H. Pendar for help with developing the mathematical model of snake flight, and F. Jafari for discussions about the role of undulation. We thank G. Nave, P. Nolan, K. Tetreault, N. Hall, F. Jafari and M. Graham for critical reading of early versions of the manuscript. We thank B. Robert, J. Settlage and A. Rizzo for help with IACUC. We also thank M. LaBarbera for his original suggestion that undulation might function as a mechanism of dynamic stability. This work was supported by the National Science Foundation under grant 1351322 to J.J.S., grants 1537349 and 1922516 to S.D.R. and grant 0966125 to S.D.R. and J.J.S.

Author information

Affiliations

Authors

Contributions

J.J.S. and I.J.Y. conceived the project, I.J.Y. and J.J.S. designed and conducted the experiments; I.J.Y. and S.D.R. developed the mathematical model; G.A.B. helped conduct experiments and export the experimental data; I.J.Y. analysed the experimental data, implemented the simulations, analysed the simulation output and produced all figures; I.J.Y., S.D.R. and J.J.S. wrote the manuscript.

Corresponding author

Correspondence to Isaac J. Yeaton.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Daniel Goldman, Michael Shelley and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

Supplementary information

41567_2020_935_MOESM3_ESM.mov

Side view of a long glide of the flying snake Chrysopelea paradisi. This slow-motion sequence shows the snake progress through the ballistic dive to the shallowing glide to landing. The sequence begins with the snake approximately 15 m above the ground, in the extended position just after a J-loop take-off (which was not recorded). During the ballistic dive, the tail is relatively up and the head is down in the pitch axis. As the snake undulates and the glide angle (relative to horizontal) decreases, the tail and posterior body move downward but continuously translate periodically in the vertical axis. Recorded in Penang, Malaysia, 2010, by producer Robert Wise, cameraman John Benam, and Jake Socha. Video provided courtesy of National Geographic Television.

41567_2020_935_MOESM4_ESM.mov

Side view of take-off and flattening of the flying snake Chrysopelea paradisi. This slow-motion sequence shows the snake begin a glide using a J-loop take-off, become dorsoventrally flattened, and then gather the body into an S-shape as it falls through the ballistic dive portion and begins undulating. The body flattening that produces the snake’s aerial cross-sectional shape starts during the jump and is completed just after the snake becomes fully airborne. Recorded in Sabah, Malaysia, 2015, by producer Simon Bell, cameraman Pete McCowen, and Jake Socha. Video provided courtesy of the British Broadcasting Corporation.

41567_2020_935_MOESM5_ESM.mov

Front view of take-off and flattening of the flying snake Chrysopelea paradisi. This slow-motion sequence shows the snake begin a glide using a J-loop take-off, become dorsoventrally flattened, and then gather the body into an S-shape as it falls through the ballistic dive portion and begins undulating. The snake appears momentarily ribbon-flat as it completes the jump, but this appearance results from the overexposure of the dorsal surface of the snake next to the bright sky. Near the end of the sequence, small bumps can be seen on the body near the tail; these bumps are sub-surface parasites that occur naturally in some wild-caught specimens. Recorded in Sabah, Malaysia, 2015, by producer Simon Bell, cameraman Pete McCowen, and Jake Socha. Video provided courtesy of the British Broadcasting Corporation.

41567_2020_935_MOESM6_ESM.mov

Overhead view of a glide trial of the flying snake Chrysopelea paradisi. Recorded with two Photron APS-RX high-speed video cameras. This glide is from a snake with a mass of 71 g and snout–vent length of 77 cm.

41567_2020_935_MOESM7_ESM.mov

Infrared markers tracked through a glide of the flying snake Chrysopelea paradisi. Top and side views are shown of 12 points on the snake as it moves through its trajectory.

41567_2020_935_MOESM8_ESM.mov

Development of a 3D model of the flying snake Chrysopelea paradisi from motion-capture data. The interactive online visualization of the snake can be found in Visualization 1: https://sketchfab.com/models/5aea787df31d48e288915b94e3a4c9de.

41567_2020_935_MOESM9_ESM.mov

Simulations of a flying snake gliding with and without aerial undulation. The co-moving frame, located at the center of mass, is indicated by the three orthogonal arrows (red, green, and blue), and the inertial coordinate system is indicated by the long lines. The instantaneous center of mass velocity is shown by the black arrow. The sheets emanating from the body are the local lift (blue) and drag (yellow) forces distributed over the body. In the first simulation, the undulating snake (Chrysopelea paradisi, f = 1.2 Hz) pitches down, but it glides 10 m vertically before any Euler angle exceeds 85°, indicating that the snake is relatively stable over short distances. In the second simulation, the snake has the same initial conditions as in the first, but with an undulation frequency of 0 Hz. The horizontal and vertical waves were selected such that the glide is generally stable in pitch. However, the snake is still unstable, as it yaws to the left more than 85° before falling 10 m vertically.

Supplementary Information

Supplementary figures, tables and text.

Reporting Summary

Supplementary Video 1

Side view of a long glide of the flying snake Chrysopelea paradisi. This slow-motion sequence shows the snake progress through the ballistic dive to the shallowing glide to landing. The sequence begins with the snake approximately 15 m above the ground, in the extended position just after a J-loop take-off (which was not recorded). During the ballistic dive, the tail is relatively up and the head is down in the pitch axis. As the snake undulates and the glide angle (relative to horizontal) decreases, the tail and posterior body move downward but continuously translate periodically in the vertical axis. Recorded in Penang, Malaysia, 2010, by producer Robert Wise, cameraman John Benam, and Jake Socha. Video provided courtesy of National Geographic Television.

Supplementary Video 2

Side view of take-off and flattening of the flying snake Chrysopelea paradisi. This slow-motion sequence shows the snake begin a glide using a J-loop take-off, become dorsoventrally flattened, and then gather the body into an S-shape as it falls through the ballistic dive portion and begins undulating. The body flattening that produces the snake’s aerial cross-sectional shape starts during the jump and is completed just after the snake becomes fully airborne. Recorded in Sabah, Malaysia, 2015, by producer Simon Bell, cameraman Pete McCowen, and Jake Socha. Video provided courtesy of the British Broadcasting Corporation.

Supplementary Video 3

Front view of take-off and flattening of the flying snake Chrysopelea paradisi. This slow-motion sequence shows the snake begin a glide using a J-loop take-off, become dorsoventrally flattened, and then gather the body into an S-shape as it falls through the ballistic dive portion and begins undulating. The snake appears momentarily ribbon-flat as it completes the jump, but this appearance results from the overexposure of the dorsal surface of the snake next to the bright sky. Near the end of the sequence, small bumps can be seen on the body near the tail; these bumps are sub-surface parasites that occur naturally in some wild-caught specimens. Recorded in Sabah, Malaysia, 2015, by producer Simon Bell, cameraman Pete McCowen, and Jake Socha. Video provided courtesy of the British Broadcasting Corporation.

Supplementary Video 4

Overhead view of a glide trial of the flying snake Chrysopelea paradisi. Recorded with two Photron APS-RX high-speed video cameras. This glide is from a snake with a mass of 71 g and snout–vent length of 77 cm.

Supplementary Video 5

Infrared markers tracked through a glide of the flying snake Chrysopelea paradisi. Top and side views are shown of 12 points on the snake as it moves through its trajectory.

Supplementary Video 6

Development of a 3D model of the flying snake Chrysopelea paradisi from motion-capture data. The interactive online visualization of the snake can be found in Visualization 1: https://sketchfab.com/models/5aea787df31d48e288915b94e3a4c9de.

Supplementary Video 7

Simulations of a flying snake gliding with and without aerial undulation. The co-moving frame, located at the center of mass, is indicated by the three orthogonal arrows (red, green, and blue), and the inertial coordinate system is indicated by the long lines. The instantaneous center of mass velocity is shown by the black arrow. The sheets emanating from the body are the local lift (blue) and drag (yellow) forces distributed over the body. In the first simulation, the undulating snake (Chrysopelea paradisi, f = 1.2 Hz) pitches down, but it glides 10 m vertically before any Euler angle exceeds 85°, indicating that the snake is relatively stable over short distances. In the second simulation, the snake has the same initial conditions as in the first, but with an undulation frequency of 0 Hz. The horizontal and vertical waves were selected such that the glide is generally stable in pitch. However, the snake is still unstable, as it yaws to the left more than 85° before falling 10 m vertically.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yeaton, I.J., Ross, S.D., Baumgardner, G.A. et al. Undulation enables gliding in flying snakes. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0935-4

Download citation