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Planar controlled gliding, tumbling and descent

Published online by Cambridge University Press:  05 December 2011

P. Paoletti  and
L. Mahadevan
P. Paoletti
Affiliation:
School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA
L. Mahadevan*
Affiliation:
School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA Department of Physics, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA
*
†Email address for correspondence: [email protected]
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Abstract

Controlled gliding during descent has been thought of as a crucial intermediate step toward the evolution of powered flight in a variety of animals. Here we develop and analyse a model for the controlled descent of thin bodies in quiescent fluids. Focusing on motion in two dimensions for simplicity, we formulate the question of steering an elliptical body to a desired landing location with a specific orientation using the framework of optimal control theory with a single control variable. We derive both time- and energy-optimal trajectories using a combination of numerical and analytical approximations. In particular, we find that energy-optimal strategies converge to constant control, while time-optimal strategies converge to bang–coast–bang control that leads to bounding flight, alternating between tumbling and gliding phases. Our study of these optimal strategies thus places natural limits on how they may be implemented in biological and biomimetic systems.

JFM classification

Flow Control: Control theory Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 489 - 516
Copyright
Copyright © Cambridge University Press 2011

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References

1. Andersen, A., Pesavento, U. & Wang, Z. J. 2005a Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91–104.CrossRefGoogle Scholar
2. Andersen, A., Pesavento, U. & Wang, Z. J. 2005b Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90.Google Scholar
3. Belmonte, A., Eisenberg, H. & Moses, E. 1998 From flutter to tumble: inertial drag and froude similarity in falling paper. Phys. Rev. Lett. 81 (2), 345–348.CrossRefGoogle Scholar
4. Benson, D. A. 2004 A Gauss pseudospectral transcription for optimal control. PhD thesis, Department of Aeronautics and Astronautics, MIT.Google Scholar
5. Benson, D. A., Huntington, G. T., Thorvaldsen, T. P. & Rao, A. V. 2006 Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method. J. Guid. Control Dyn. 29 (6), 1435–1440.CrossRefGoogle Scholar
6. Bradley, T. J., Briscoe, A. D., Brady, S. G., Contreras, H. L., Danforth, B. N., Dudley, R., Grimaldi, D., Harrison, J. F., Kaiser, J. A., Merlin, C., Reppert, S. M., VandenBrooks, J. M. & Yanoviak, S. P. 2009 Episodes in insect evolution. Integr. Compar. Biol. 49 (5), 590–606.CrossRefGoogle ScholarPubMed
7. Cory, R. & Tedrake, R. 2008 Experiments in fixed-wing UAV perching. In AIAA Guidance, Navigation, and Control Conference.Google Scholar
8. Dudley, R. 2000 The Biomechanics of Insect Flight: Form, Function, Evolution. Princeton University Press.CrossRefGoogle Scholar
9. Dudley, R., Byrnes, G., Yanoviak, S. P., Borrell, B., Brown, R. M. & McGuire, J. A. 2007 Gliding and the functional origins of flight: biomechanical novelty or necessity? Annu. Rev. Ecol. Evol. Systemat. 38 (1), 179–201.CrossRefGoogle Scholar
10. Garg, D., Patterson, M. A., Hager, W. W., Rao, A. V., Benson, D. A. & Huntington, G. T. 2010 A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46 (11), 1843–1851.CrossRefGoogle Scholar
11. Gill, P. E., Murray, W. & Saunders, M. A. 2005 SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47 (1), 99–131.CrossRefGoogle Scholar
12. Grimaldi, D. A. & Engel, M. S. 2005 Evolution of the Insects. Cambridge University Press.Google Scholar
13. Hasenfuss, I. 2008 The evolutionary pathway to insect flight – a tentative reconstruction. Arthropod Systemat. Phylogeny 66 (1), 19–35.Google Scholar
14. Huntington, G. T. 2007 Advancement and analysis of a gauss pseudospectral transcription for optimal control. PhD thesis, Dept. of Aeronautics and Astronautics, MIT.Google Scholar
15. Huntington, G. T., Benson, D. A. & Rao, A. V. 2007a Design of optimal tetrahedral spacecraft formations. J. Astronautical Sci. 55 (2), 141–169.Google Scholar
16. Huntington, G. T., Benson, D. A., How, J. P., Kanizay, N., Darby, C. L. & Rao, A. V. 2007 b Computation of boundary controls using a gauss pseudospectral method. In 2007 Astrodynamics Specialist Conference, Mackinac Island, Michigan.Google Scholar
17. Huntington, G. T. & Rao, A. V. 2008 Optimal reconfiguration of spacecraft formations using a gauss pseudospectral method. J. Guid. Control Dyn. 31 (3), 689–698.Google Scholar
18. Jackson, S. M. 2000 Glide angle in the genus petaurus and a review of gliding in mammals. Mammal Rev. 30 (1), 9–30.CrossRefGoogle Scholar
19. Kirk, D. E. 2004 Optimal Control Theory: An Introduction. Dover.Google Scholar
20. Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
21. Mahadevan, L. 1996 Tumbling of a falling card. C. R. Acad. Sci. Sér. II 323, 729–736.Google Scholar
22. Mahadevan, L., Ryu, W. S. & Aravinthan, D. T. S. 1999 Tumbling cards. Phys. Fluids 11, 1–3.CrossRefGoogle Scholar
23. Mittal, R., Seshadri, V. & Udaykumar, H. S. 2004 Flutter, tumble and vortex induced autorotation. Theor. Comput. Fluid Dyn. 17 (3), 165–170.CrossRefGoogle Scholar
24. Pesavento, U. & Wang, Z. J. 2004 Falling paper: Navier–Stokes solutions, model of fluid forces, and centre of mass elevation. Phys. Rev. Lett. 93 (14), 144501.CrossRefGoogle ScholarPubMed
25. Pontryagin, L. S, Boltyanskii, V. G., Gamkrelidze, R. V. & Mischenko, E. F. 1962 The Mathematical Theory of Optimal Processes. Wiley-Interscience.Google Scholar
26. Rao, A. V., Benson, D. A., Darby, C. L., Patterson, M. A., Francolin, C. & Huntington, G. T. 2010 Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method. ACM Trans. Math. Softw. 37 (2), 22:1–39.Google Scholar
27. Rayner, J. M. V. 1985 Bounding and undulating flight in birds. J. Theor. Biol. 117, 47–77.Google Scholar
28. Roberts, John W., Cory, Rick & Tedrake, Russ 2009 On the controllability of fixed-wing perching. In Proceedings of the American Controls Conference (ACC).Google Scholar
29. Tobalske, B. W. 2010 Hovering and intermittent flight in birds. Bioinspiration Biomimetics 5 (4), 045004.Google Scholar
30. Wang, Z. J., Birch, J. M. & Dickinson, M. H. 2004 Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments. J. Expl. Biol. 207 (3), 449–460.Google Scholar
31. Woolsey, C. A. & Leonard, N. E. 2002 Moving mass control for underwater vehicles. Proceedings of the 2002 American Control Conference, vol. 4, pp. 2824–2829.Google Scholar
32. Yanoviak, S. P., Dudley, R. & Kaspari, M. 2005 Directed areal descent in canopy ants. Nature 433, 624–626.Google Scholar
33. Yanoviak, S., Fisher, B. & Alonso, A. 2008 Directed Aerial Descent Behavior in African Canopy Ants (Hymenoptera: Formicidae). J. Insect Behavior 21 (3), 164–171.CrossRefGoogle Scholar
34. Yanoviak, S. P, Kaspari, M. & Dudley, R. 2009 Gliding hexapods and the origins of insect aerial behaviour. Biol. Lett. 5 (4), 510–512.CrossRefGoogle ScholarPubMed
35. Yanoviak, S. P., Munk, Y., Kaspari, M. & Dudley, R. 2010 Aerial manoeuvrability in wingless gliding ants (Cephalotes atratus). Proc. R. Soc. B Biol. Sci. 277 (1691), 2199–2204.Google Scholar