- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Smart navigation of a gravity-driven glider with adjustable center of mass

X. Jiang1,*, J. Qiu2,*, K. Gustavsson2, B. Mehlig2, and L. Zhao1,3,†

  • *These authors contributed equally to this work.
  • †Contact author: zhaolihao@tsinghua.edu.cn
PDF

Phys. Rev. Research 7, 043200 – Published 21 November, 2025

DOI: https://doi.org/10.1103/1r5c-qpng

Abstract

Artificial gliders are designed to disperse as they settle through a fluid, requiring precise navigation to reach target locations. We show that a compact glider settling in a viscous fluid can navigate by dynamically adjusting its center of mass. Using fully resolved direct numerical simulations and reinforcement learning, we find two optimal navigation strategies that allow the glider to reach its target location accurately. These strategies depend sensitively on how the glider interacts with the surrounding fluid. The nature of this interaction changes as the particle Reynolds number Rep changes. Our results explain how the optimal strategy depends on Rep. At large Rep, the glider learns to tumble rapidly by moving its center of mass as its orientation changes. This generates a large horizontal inertial lift force, which allows the glider to travel far. At small Rep, by contrast, high viscosity hinders tumbling. In this case, the glider learns to adjust its center of mass so that it settles with a steady, inclined orientation that results in a horizontal viscous force. The horizontal range is much smaller than for large Rep, because this viscous force is much smaller than the inertial lift force at large Rep.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (49)

  1. S. P. Yanoviak, R. Dudley, and M. Kaspari, Directed aerial descent in canopy ants, Nature (London) 433, 624 (2005).
  2. C. Cummins, M. Seale, A. Macente, D. Certini, E. Mastropaolo, I. M. Viola, and N. Nakayama, A separated vortex ring underlies the flight of the dandelion, Nature (London) 562, 414 (2018).
  3. C. K. Augspurger, Morphology and dispersal potential of wind-dispersed diaspores of neotropical trees, Am. J. Bot. 73, 353 (1986).
  4. B. H. Kim, K. Li, J. Kim, Y. Park, H. Jang et al., Three-dimensional electronic microfliers inspired by wind-dispersed seeds, Nature (London) 597, 503 (2021).
  5. J. M. Kahn, R. H. Katz, and K. S. J. Pister, Next century challenges: Mobile networking for smart dust, in Proceedings of the 5th Annual ACM/IEEE International Conference on Mobile Computing and Networking (ACM, Seattle, WA, 1999), pp. 271–278.
  6. P. F. J. Lermusiaux, D. N. Subramani, J. Lin, C. S. Kulkarni, A. Gupta, A. Dutt, T. Lolla, P. J. Haley, Jr., W. H. Ali, C. Mirabito, and S. Jana, A future for intelligent autonomous ocean observing systems, J. Mar. Res. 75, 765 (2017).
  7. N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M Fratantoni, and R. E Davis, Collective motion, sensor networks, and ocean sampling, Proc. IEEE 95, 48 (2007).
  8. B. Mitchell, E. Wilkening, and N. Mahmoudian, Low cost underwater gliders for littoral marine research, in American Control Conference (ACC), Proceedings of the American Control Conference (IEEE, Washington, DC, USA, 2013), pp. 1412–1417.
  9. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Springer Science & Business Media, Berlin, Germany, 1983).
  10. J. Pierson, M. Kharrouba, and J. Magnaudet, Hydrodynamic torque on a slender cylinder rotating perpendicularly to its symmetry axis, Phys. Rev. Fluids 6, 094303 (2021).
  11. R. E. Khayat and R. G. Cox, Inertia effects on the motion of long slender bodies, J. Fluid Mech. 209, 435 (1989).
  12. D. Lopez and E. Guazzelli, Inertial effects on fibers settling in a vortical flow, Phys. Rev. Fluids 2, 024306 (2017).
  13. K. Gustavsson, M. Z. Sheikh, A. Naso, A. Pumir, and B. Mehlig, Effect of particle inertia on the alignment of small ice crystals in turbulent clouds, J. Atmos. Sci. 78, 2573 (2021).
  14. F. Candelier, R. Mehaddi, B. Mehlig, and J. Magnaudet, Second-order inertial forces and torques on a sphere in a viscous steady linear flow, J. Fluid Mech. 954, A25 (2023).
  15. P. Ern, F. Risso, D. Fabre, and J. Magnaudet, Wake-induced oscillatory paths of bodies freely rising or falling in fluids, Annu. Rev. Fluid Mech. 44, 97 (2012).
  16. U. Pesavento and Z. J. Wang, Falling paper Navier-Stokes solutions model of fluid forces and center of mass elevation, Phys. Rev. Lett. 93, 144501 (2004).
  17. A. Andersen, U. Pesavento, and Z. J. Wang, Analysis of transitions between fluttering, tumbling and steady descent of falling cards, J. Fluid Mech. 541, 91 (2005).
  18. S. B. Field, M. Klaus, M. G. Moore, and F. Nori, Chaotic dynamics of falling disks, Nature (London) 388, 252 (1997).
  19. P. Paoletti and L. Mahadevan, Planar controlled gliding, tumbling and descent, J. Fluid Mech. 689, 489 (2011).
  20. V. Mnih, K. Kavukcuoglu, D. Silver, Andrei A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, Human-level control through deep reinforcement learning, Nature (London) 518, 529 (2015).
  21. B. Mehlig, Machine Learning with Neural Networks: An Introduction for Scientists and Engineers (Cambridge University Press, Cambridge, 2021).
  22. S. Colabrese, K. Gustavsson, A. Celani, and L. Biferale, Flow navigation by smart microswimmers via reinforcement learning, Phys. Rev. Lett. 118, 158004 (2017).
  23. G. Novati, L. Mahadevan, and P. Koumoutsakos, Controlled gliding and perching through deep-reinforcement-learning, Phys. Rev. Fluids 4, 093902 (2019).
  24. P. Gunnarson, I. Mandralis, G. Novati, P. Koumoutsakos, and J. O. Dabiri, Learning efficient navigation in vortical flow fields, Nat. Commun. 12, 7143 (2021).
  25. J. K. Alageshan, A. K. Verma, J. Bec, and R. Pandit, Machine learning strategies for path-planning microswimmers in turbulent flows, Phys. Rev. E 101, 043110 (2020).
  26. J. Jiang, Z. Yang, A. Ferreira, and L. Zhang, Control and autonomy of microrobots: Recent progress and perspective, Adv. Intell. Syst. 4, 2100279 (2022).
  27. A. Roy, R. J. Hamati, L. Tierney, D. L. Koch, and G. A. Voth, Inertial torques and a symmetry breaking orientational transition in the sedimentation of slender fibres, J. Fluid Mech. 875, 576 (2019).
  28. X. Jiang, C. Xu, and L. Zhao, Settling and collision of spheroidal particles with an offset mass centre in a quiescent fluid, J. Fluid Mech. 984, A40 (2024).
  29. T. Bhowmick, J. Seesing, K. Gustavsson, J. Guettler, Y. Wang, A. Pumir, B. Mehlig, and G. Bagheri, Inertia induces strong orientation fluctuations of nonspherical atmospheric particles, Phys. Rev. Lett. 132, 034101 (2024).
  30. C. S. Peskin, The immersed boundary method, Acta Numer. 11, 479 (2002).
  31. M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys. 209, 448 (2005).
  32. W. Breugem, A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows, J. Comput. Phys. 231, 4469 (2012).
  33. K. Kim, S. J. Baek, and H. J. Sung, An implicit velocity decoupling procedure for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 38, 125 (2002).
  34. X. Jiang, W. Huang, C. Xu, and L. Zhao, A flow-reconstruction based approach for the computation of hydrodynamic stresses on immersed body surface, J. Comput. Phys. 508, 113025 (2024).
  35. H. van Hasselt, A. Guez, and D. Silver, Deep reinforcement learning with double Q-learning, Proc. AAAI Conf. Artif. Intell. 30, 2094 (2016).
  36. R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction, 2nd ed. (The MIT Press, Cambridge, Massachusetts, USA, 2018).
  37. H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, United Kingdom, 1924).
  38. L. I. Sedov, Two-Dimensional Problems of Hydrodynamics and Aerodynamics (Izdatel Nauka, Moscow, 1980).
  39. G. Oh, H. Park, and J. Choi, Drag, lift, and torque coefficients for various geometrical configurations of elliptic cylinder under Stokes to Laminar flow regimes, AIP Adv. 12, 065228 (2022).
  40. S. I. Rubinow and Joseph B. Keller, The transverse force on a spinning sphere moving in a viscous fluid, J. Fluid Mech. 11, 447 (1961).
  41. J. S. Jaffe, P. J. S. Franks, P. L. D. Roberts, D. Mirza, C. Schurgers, R. Kastner, and A. Boch, A swarm of autonomous miniature underwater robot drifters for exploring submesoscale ocean dynamics, Nat. Commun. 8, 14189 (2017).
  42. InvenSense, MPU-6000 and MPU-6050 register map and descriptions revision 4.2, 2013, retrieved from InvenSense website, https://invensense.tdk.com/wp-content/uploads/2015/02/MPU-6000-Datasheet1.pdf.
  43. C. Harvey, V. B. Baliga, P. Lavoie, and D. L. Altshuler, Wing morphing allows gulls to modulate static pitch stability during gliding, J. R. Soc. Interface 16, 20180641 (2019).
  44. F. Candelier, K. Gustavsson, P. Sharma, L. Sundberg, A. Pumir, G. Bagheri, and B. Mehlig, Torques on curved atmospheric fibers, Phys. Rev. Res. 7, 013179 (2025).
  45. E. Huseby, J. Gissinger, F. Candelier, N. Pujara, G. Verhille, B. Mehlig, and G. Voth, Helical ribbons: Simple chiral sedimentation, Phys. Rev. Fluids 10, 024101 (2025).
  46. F. Candelier and B. Mehlig, Settling of an asymmetric dumbbell in a quiescent fluid, J. Fluid Mech. 802, 174 (2016).
  47. A. Roy, S. Kramel, U. Menon, G. A. Voth, and D. L. Koch, Orientation of finite Reynolds number anisotropic particles settling in turbulence, J. Non-Newtonian Fluid Mech. 318, 105048 (2023).
  48. S. Ravichandran and J. S. Wettlaufer, Orientation dynamics of two-dimensional concavo-convex bodies, Phys. Rev. Fluids 8, L062301 (2023).
  49. Z. Maches, M. Houssais, A. Sauret, and E. Meiburg, Settling of two rigidly connected spheres, J. Fluid Mech. 995, A15 (2024).

Outline

Information

Phys. Rev. Research 7, 043200– Published 21 November, 2025
  • Received 1 July 2024
  • Accepted 19 October 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation