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Nonlinear tunable stiffness for high-efficiency biomimetic propulsion

Published online by Cambridge University Press:  11 August 2025

Yiming Lu ,
Haicheng Zhang Open the ORCID record for Haicheng Zhang [Opens in a new window] ,
Daolin Xu  and
Wei-Xi Huang Open the ORCID record for Wei-Xi Huang [Opens in a new window]
Yiming Lu
Affiliation:
College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China
Haicheng Zhang*
Affiliation:
College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China Greater Bay Area Institute for Innovation, Hunan University, Guangzhou 511300, PR China Wuxi Intelligent Control Research Institute, Hunan University, Wuxi 214082, PR China
Daolin Xu
Affiliation:
College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China
Wei-Xi Huang*
Affiliation:
AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
*
Corresponding authors: Haicheng Zhang, zhanghc@hnu.edu.cn; Wei-Xi Huang, hwx@tsinghua.edu.cn
Corresponding authors: Haicheng Zhang, zhanghc@hnu.edu.cn; Wei-Xi Huang, hwx@tsinghua.edu.cn
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Abstract

Several million years of natural evolution have endowed marine animals with high flexibility and mobility. A key factor in this achievement is their ability to modulate stiffness during swimming. However, an unresolved puzzle remains regarding how muscles modulate stiffness, and the implications of this capability for achieving high swimming efficiency. Inspired by this, we proposed a self-propulsor model that employs a parabolic stiffness-tuning strategy, emulating the muscle tensioning observed in biological counterparts. Furthermore, efforts have been directed towards developing the nonlinear vortex sheet method, specifically designed to address nonlinear fluid–structure coupling problems. This work aims to analyse how and why nonlinear tunable stiffness influences swimming performance. Numerical results demonstrate that swimmers with nonlinear tunable stiffness can double their speed and efficiency across nearly the entire frequency range. Additionally, our findings reveal that high-efficiency biomimetic propulsion originates from snap-through instability, which facilitates the emergence of quasi-quadrilateral swimming patterns and enhances vortex strength. Moreover, this study examines the influence of nonlinear stiffness on swimming performance, providing valuable insights into the optimisation of next-generation, high-performance, fish-inspired robotic systems.

JFM classification

Biological Fluid Dynamics: Propulsion Vortex Flows: Vortex dynamics Mathematical Foundations: Computational methods

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JFM Papers
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Journal of Fluid Mechanics , Volume 1017 , 25 August 2025 , A7
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© The Author(s), 2025. Published by Cambridge University Press

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References

Abu Zeid, H. & Elshahawi, H. 2023 Advances in ROV & AUV Capabilities and Tooling for Energy Applications. OnePetro.10.4043/32298-MSCrossRefGoogle Scholar
Alben, S. 2008 Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.10.1017/S0022112008003297CrossRefGoogle Scholar
Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 2587–2603.10.1016/j.jcp.2008.12.020CrossRefGoogle Scholar
Alben, S., Madden, P.G. & Lauder, G.V. 2007 The mechanics of active fin-shape control in ray-finned fishes. J. R. Soc. Interface 4 (13), 243–256.10.1098/rsif.2006.0181CrossRefGoogle ScholarPubMed
Altringham, J.D. & Ellerby, D.J. 1999 Fish swimming: patterns in muscle function. J. Expl Biol. 202 (23), 3397–3403.10.1242/jeb.202.23.3397CrossRefGoogle ScholarPubMed
Anderson, J.M., Streitlien, K., Barrett, D.S. & Triantafyllou, M.S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72.10.1017/S0022112097008392CrossRefGoogle Scholar
Arora, N., Kang, C.-K., Shyy, W. & Gupta, A. 2018 Analysis of passive flexion in propelling a plunging plate using a torsion spring model. J. Fluid Mech. 857, 562–604.10.1017/jfm.2018.736CrossRefGoogle Scholar
Boyer, F., Porez, M., Leroyer, A. & Visonneau, M. 2008 Fast dynamics of an eel-like robot – comparisons with Navier–Stokes simulations. IEEE Trans. Robot. 24 (6), 1274–1288.10.1109/TRO.2008.2006249CrossRefGoogle Scholar
Broyden, C.G. 1965 A class of methods for solving nonlinear simultaneous equations. Maths Comput. 19 (92), 577–593.10.1090/S0025-5718-1965-0198670-6CrossRefGoogle Scholar
Chen, D., Wu, Z., Dong, H., Tan, M. & Yu, J. 2020 Exploration of swimming performance for a biomimetic multi-joint robotic fish with a compliant passive joint. Bioinspir. Biomim. 16 (2), 026007.10.1088/1748-3190/abc494CrossRefGoogle ScholarPubMed
Chen, L., Liao, X., Xia, G., Sun, B. & Zhou, Y. 2023 Variable-potential bistable nonlinear energy sink for enhanced vibration suppression and energy harvesting. Intl J. Mech. Sci. 242, 107997.10.1016/j.ijmecsci.2022.107997CrossRefGoogle Scholar
Chi, Y., Hong, Y., Zhao, Y., Li, Y. & Yin, J. 2022 a Snapping for high-speed and high-efficient butterfly stroke-like soft swimmer. Sci. Adv. 8 (46), eadd3788.10.1126/sciadv.add3788CrossRefGoogle ScholarPubMed
Chi, Y., Li, Y., Zhao, Y., Hong, Y., Tang, Y. & Yin, J. 2022 b Bistable and multistable actuators for soft robots: structures, materials, and functionalities. Adv. Mater. 34 (19), 2110384.10.1002/adma.202110384CrossRefGoogle ScholarPubMed
Chopra, M.G. 1974 Hydromechanics of lunate-tail swimming propulsion. J. Fluid Mech. 64 (2), 375–392.10.1017/S002211207400245XCrossRefGoogle Scholar
Crank, J. & Nicolson, P. 1947 A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type. pp. 50–67. Cambridge University Press.Google Scholar
Dickinson, M.H., Farley, C.T., Full, R.J., Koehl, M.A.R., Kram, R. & Lehman, S. 2000 How animals move: an integrative view. Science 288, 100–106.10.1126/science.288.5463.100CrossRefGoogle ScholarPubMed
Drucker, E.G. & Lauder, G.V. 2001 Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish. J. Expl Biol. 204 (17), 2943–2958.10.1242/jeb.204.17.2943CrossRefGoogle ScholarPubMed
Duraisamy, P., Kumar Sidharthan, R. & Nagarajan Santhanakrishnan, M. 2019 Design, modeling, and control of biomimetic fish robot: a review. J. Bionic Engng 16, 967–993.10.1007/s42235-019-0111-7CrossRefGoogle Scholar
El-Atab, N., Mishra, R.B., Al-Modaf, F., Joharji, L., Alsharif, A.A., Alamoudi, H., Diaz, M., Qaiser, N. & Hussain, M.M. 2020 Soft actuators for soft robotic applications: a review. Adv. Intell. Syst. 2 (10), 2000128.10.1002/aisy.202000128CrossRefGoogle Scholar
Esposito, C.J., Tangorra, J.L., Flammang, B.E. & Lauder, G.V. 2012 A robotic fish caudal fin: effects of stiffness and motor program on locomotor performance. J. Expl Biol. 215 (1), 56–67.10.1242/jeb.062711CrossRefGoogle ScholarPubMed
Fang, F., Ho, K.L., Ristroph, L. & Shelley, M.J. 2017 A computational model of the flight dynamics and aerodynamics of a jellyfish-like flying machine. J. Fluid Mech. 819, 621–655.10.1017/jfm.2017.150CrossRefGoogle Scholar
Flammang, B.E. & Lauder, G.V. 2008 Speed-dependent intrinsic caudal fin muscle recruitment during steady swimming in bluegill sunfish, Lepomis macrochirus . J. Expl. Biol. 211 (4), 587–598.10.1242/jeb.012096CrossRefGoogle ScholarPubMed
Hang, H., Heydari, S., Costello, J.H. & Kanso, E. 2022 Active tail flexion in concert with passive hydrodynamic forces improves swimming speed and efficiency. J. Fluid Mech. 932, A35.10.1017/jfm.2021.984CrossRefGoogle Scholar
Harper, D.G. & Blake, R.W. 1990 Fast-start performance of rainbow trout Salmo gairdneri and northern pike Esox lucius . J. Exp. Biol. 150 (1), 321–342.10.1242/jeb.150.1.321CrossRefGoogle Scholar
Hartmann, F., Baskaran, M., Raynaud, G., Benbedda, M., Mulleners, K. & Shea, H. 2025 Highly agile flat swimming robot. Science Robotics 10 (99), eadr0721.10.1126/scirobotics.adr0721CrossRefGoogle ScholarPubMed
He, C. & Zhao, J. 2009 Modeling rotor wake dynamics with viscous vortex particle method. AIAA J. 47 (4), 902–915.10.2514/1.36466CrossRefGoogle Scholar
Heydari, S. & Kanso, E. 2021 School cohesion, speed and efficiency are modulated by the swimmers flapping motion. J. Fluid Mech. 922, A27.10.1017/jfm.2021.551CrossRefGoogle Scholar
Hu, H., Liu, J., Dukes, I. & Francis, G. 2006 Design of 3D swim patterns for autonomous robotic fish. In 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2406–2411. IEEE.10.1109/IROS.2006.281680CrossRefGoogle Scholar
Huang, Y., Nitsche, M. & Kanso, E. 2016 Hovering in oscillatory flows. J. Fluid Mech. 804, 531–549.10.1017/jfm.2016.535CrossRefGoogle Scholar
Huang, Y., Ristroph, L., Luhar, M. & Kanso, E. 2018 Bistability in the rotational motion of rigid and flexible flyers. J. Fluid Mech. 849, 1043–1067.10.1017/jfm.2018.446CrossRefGoogle Scholar
Jayne, B.C. & Lauder, G.V. 1995 Speed effects on midline kinematics during steady undulatory swimming of largemouth bass, Micropterus salmoides . J. Exp. Biol. 198 (2), 585–602.10.1242/jeb.198.2.585CrossRefGoogle Scholar
Jin, H., Zhang, H., Zheng, S., Lu, Y., Xu, D. & Greaves, D. 2024 Dual-purpose wave farm with nonlinear stiffness mechanism for energy extraction and wave attenuation. Phys. Fluids 36 (9), 097126.10.1063/5.0227165CrossRefGoogle Scholar
Jones, M.A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.10.1017/S0022112003006645CrossRefGoogle Scholar
Jusufi, A., Vogt, D.M., Wood, R.J. & Lauder, G.V. 2017 Undulatory swimming performance and body stiffness modulation in a soft robotic fish-inspired physical model. Soft Robot. 4 (3), 202–210.10.1089/soro.2016.0053CrossRefGoogle Scholar
Katz, J. & Weihs, D. 1978 Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech. 88 (3), 485–497.10.1017/S0022112078002220CrossRefGoogle Scholar
Lane, D.M., Sfakiotakis, M. & Davies, B.J. 1998 Review of fish swimming modes for aquatic locomotion. IEEE J. Oceanic Engng 24–22.10.1109/48.757275CrossRefGoogle Scholar
Lauder, G.V., Madden, P.G.A., Tangorra, J.L., Anderson, E. & Baker, T.V. 2011 Bioinspiration from fish for smart material design and function. Smart Mater. Struct. 20 (9), 094014.10.1088/0964-1726/20/9/094014CrossRefGoogle Scholar
Li, G. 2021 Self-powered soft robot in the Mariana Trench. Nature 591, 66–71.10.1038/s41586-020-03153-zCrossRefGoogle ScholarPubMed
Li, K., Jiang, H., Wang, S. & Yu, J. 2018 a A soft robotic fish with variable-stiffness decoupled mechanisms. J. Bionic Engng 15, 599–609,10.1007/s42235-018-0049-1CrossRefGoogle Scholar
Li, R., Xiao, Q., Liu, Y., Hu, J., Li, L., Li, G., Liu, H., Hu, K. & Wen, L. 2018 b A multi-body dynamics based numerical modelling tool for solving aquatic biomimetic problems. Bioinspir. Biomim. 13 (5), 056001,10.1088/1748-3190/aacd60CrossRefGoogle ScholarPubMed
Li, Z.-Y., Feng, L.-H., Wang, T. & Liang, Y. 2023 Lift generation mechanism of the leading-edge vortex for an unsteady plate. J. Fluid Mech. 972, A30.10.1017/jfm.2023.569CrossRefGoogle Scholar
Liao, J.C., Beal, D.N., Lauder, G.V. & Triantafyllou, M.S. 2003 Fish exploiting vortices decrease muscle activity. Science 302, 1566–1569.10.1126/science.1088295CrossRefGoogle ScholarPubMed
Lighthill, M.J. 1969 Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1 (1), 413–446.10.1146/annurev.fl.01.010169.002213CrossRefGoogle Scholar
Liu, C., Zhang, W., Yu, K., Liu, T. & Zheng, Y. 2024 Quasi-zero-stiffness vibration isolation: designs, improvements and applications. Engng Struct. 301, 117282.10.1016/j.engstruct.2023.117282CrossRefGoogle Scholar
Liu, G., Ren, Y., Dong, H., Akanyeti, O., Liao, J.C. & Lauder, G.V. 2017 Computational analysis of vortex dynamics and performance enhancement due to body–fin and fin–fin interactions in fish-like locomotion. J. Fluid Mech. 829, 65–88.10.1017/jfm.2017.533CrossRefGoogle Scholar
Liu, J. & Hu, H. 2007 A methodology of modelling fish-like swim patterns for robotic fish. In Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, pp. 1316–1321. IEEE.10.1109/ICMA.2007.4303740CrossRefGoogle Scholar
Liu, S., Liu, C., Liang, Y., Ren, L. & Ren, L. 2023 Tunable stiffness caudal peduncle leads to higher swimming speed without extra energy. IEEE Robot. Automat. Lett. 8 (9), 5886–5893.10.1109/LRA.2023.3300587CrossRefGoogle Scholar
Long, Jr & John, H. 1998 Muscles, elastic energy, and the dynamics of body stiffness in swimming eels. Am. Zool. 38 (4), 771–792.10.1093/icb/38.4.771CrossRefGoogle Scholar
Mason, R. & Burdick, J.W. 2000 Experiments in carangiform robotic fish locomotion. In Proceedings of the 2000 IEEE International Conference on Robotics and Automation, vol. 1, pp. 428–435. IEEE.10.1109/ROBOT.2000.844093CrossRefGoogle Scholar
Matthews, D.G., Zhu, R., Wang, J., Dong, H., Bart-Smith, H. & Lauder, G. 2022 Role of the caudal peduncle in a fish-inspired robotic model: how changing stiffness and angle of attack affects swimming performance. Bioinspir. Biomim. 17 (6), 066017.10.1088/1748-3190/ac9879CrossRefGoogle Scholar
Michelin, S., Smith, S.G.L. & Glover, B.J. 2008 Vortex shedding model of a flapping flag. J. Fluid Mech. 617, 1–10.10.1017/S0022112008004321CrossRefGoogle Scholar
Nakabayashi, M., Kobayashi, R., Kobayashi, S. & Morikawa, H. 2009 Bioinspired propulsion mechanism using a fin with a dynamic variable-effective-length spring-evaluation of thrust characteristics and flow around a fin in a uniform flow. J. Biomech. Sci. Engng 4 (1), 82–93.10.1299/jbse.4.82CrossRefGoogle Scholar
Ni, X., Liao, C., Li, Y., Zhang, Z., Sun, M., Chai, H., Wu, H. & Jiang, S. 2020 Experimental study of multi-stable morphing structures actuated by pneumatic actuation. Intl J. Adv. Manufacturing Technol. 108, 1203–1216.10.1007/s00170-020-05301-1CrossRefGoogle Scholar
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161.10.1017/S0022112094002508CrossRefGoogle Scholar
Overvelde, J.T.B., Kloek, T., D’haen, J.J.A. & Bertoldi, K. 2015 Amplifying the response of soft actuators by harnessing snap-through instabilities. Proc. Natl Acad. Sci. USA 112 (35), 10863–10868.10.1073/pnas.1504947112CrossRefGoogle ScholarPubMed
Pabst, D.A. 1996 Springs in swimming animals. Am. Zool. 36 (6), 723–735.10.1093/icb/36.6.723CrossRefGoogle Scholar
Pal, A., Restrepo, V., Goswami, D. & Martinez, R.V. 2021 Exploiting mechanical instabilities in soft robotics: control, sensing, and actuation. Adv. Mater. 33 (19), 2006939.10.1002/adma.202006939CrossRefGoogle ScholarPubMed
Pan, F. 2025 Miniature deep-sea morphable robot with multimodal locomotion. Sci. Robot. 10 (100), eadp7821.10.1126/scirobotics.adp7821CrossRefGoogle ScholarPubMed
Park, Y.-J., Huh, T.M., Park, D. & Cho, K.-J. 2014 Design of a variable-stiffness flapping mechanism for maximizing the thrust of a bio-inspired underwater robot. Bioinspir. Biomim. 9 (3), 036002.10.1088/1748-3182/9/3/036002CrossRefGoogle ScholarPubMed
Peng, Z.-R., Sun, Y., Yang, D., Xiong, Y., Wang, L. & Wang, L. 2022 Scaling laws for drag-to-thrust transition and propulsive performance in pitching flexible plates. J. Fluid Mech. 941, R2.10.1017/jfm.2022.268CrossRefGoogle Scholar
Quinn, D. & Lauder, G. 2021 Tunable stiffness in fish robotics: mechanisms and advantages. Bioinspir. Biomim. 17 (1), 011002.10.1088/1748-3190/ac3ca5CrossRefGoogle ScholarPubMed
Quinn, D.B., Lauder, G.V. & Smits, A.J. 2015 Maximizing the efficiency of a flexible propulsor using experimental optimization. J. Fluid Mech. 767, 430–448.10.1017/jfm.2015.35CrossRefGoogle Scholar
Ramananarivo, S., Fang, F., Oza, A., Zhang, J. & Ristroph, L. 2016 Flow interactions lead to orderly formations of flapping wings in forward flight. Phys. Rev. Fluids 1 (7), 071201.10.1103/PhysRevFluids.1.071201CrossRefGoogle Scholar
Ren, Z., Yang, X., Wang, T. & Wen, L. 2016 Hydrodynamics of a robotic fish tail: effects of the caudal peduncle, fin ray motions and the flow speed. Bioinspir. Biomim. 11 (1), 016008.10.1088/1748-3190/11/1/016008CrossRefGoogle ScholarPubMed
Renda, F., Giorgio-Serchi, F., Boyer, F., Laschi, C., Dias, J. & Seneviratne, L. 2018 A unified multi-soft-body dynamic model for underwater soft robots. Intl J. Robot. Res. 37 (6), 648–666.10.1177/0278364918769992CrossRefGoogle Scholar
Sarpkaya, T. 1975 An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate. J. Fluid Mech. 68 (1), 109–128.10.1017/S0022112075000717CrossRefGoogle Scholar
Shahria, T., Zaman, K.T., Rabbi, S. & Khan, M.M. 2019 Underwater research and rescue robot. In 2019 IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT), pp. 1–5, IEEE.10.1109/ICECCT.2019.8869287CrossRefGoogle Scholar
Shao, C., Chi, J., Zhang, H., Fan, Q., Zhao, Y. & Ye, F. 2020 Development of cell spheroids by advanced technologies. Adv. Materials Technol. 5 (9), 2000183.10.1002/admt.202000183CrossRefGoogle Scholar
Shelley, M.J. 1992 A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244, 493–526.10.1017/S0022112092003161CrossRefGoogle Scholar
Ta, T.D., Umedachi, T. & Kawahara, Y. 2020 A multigait stringy robot with bi-stable soft-bodied structures in multiple viscous environments. In 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 8765–8772. IEEE.10.1109/IROS45743.2020.9341059CrossRefGoogle Scholar
Taneda, S. 1965 Experimental investigation of vortex streets. J. Phys. Soc. Japan 20 (9), 1714–1721.10.1143/JPSJ.20.1714CrossRefGoogle Scholar
Triantafyllou, G.S. & Triantafyllou, M.S. 1993 Reverse Kármán streets in the wake of flapping foils for application in fish propulsion. In Bluff-Body Wakes, Dynamics and Instabilities, pp. 11–14. Springer.10.1007/978-3-662-00414-2_2CrossRefGoogle Scholar
Triantafyllou, M.S. & Triantafyllou, G.S. 1995 An efficient swimming machine. Sci. Am. 272 (3), 64–70.10.1038/scientificamerican0395-64CrossRefGoogle Scholar
Tytell, E.D., Leftwich, M.C., Hsu, C.-Y., Griffith, B.E., Cohen, A.H., Smits, A.J., Hamlet, C. & Fauci, L.J. 2016 Role of body stiffness in undulatory swimming: insights from robotic and computational models. Phys. Rev. Fluids 1 (7), 073202.10.1103/PhysRevFluids.1.073202CrossRefGoogle Scholar
Walker, J.A. & Westneat, M.W. 2002 Kinematics, dynamics, and energetics of rowing and flapping propulsion in fishes. Integr. Comp. Biol. 42 (5), 1032–1043.10.1093/icb/42.5.1032CrossRefGoogle ScholarPubMed
Wang, T., Ren, Z., Hu, W., Li, M. & Sitti, M. 2021 Effect of body stiffness distribution on larval fish-like efficient undulatory swimming. Sci. Adv. 7 (19), eabf7364.10.1126/sciadv.abf7364CrossRefGoogle ScholarPubMed
Webb, P.W. 1984 Form and function in fish swimming. Sci. Am. 251 (1), 72–83.10.1038/scientificamerican0784-72CrossRefGoogle Scholar
Wei, T., Mark, R. & Hutchison, S. 2014 The fluid dynamics of competitive swimming. Annu. Rev. Fluid Mech. 46 (1), 547–565.10.1146/annurev-fluid-011212-140658CrossRefGoogle Scholar
White, C.H., Lauder, G.V. & Bart-Smith, H. 2021 Tunabot flex: a tuna-inspired robot with body flexibility improves high-performance swimming. Bioinspir. Biomim. 16 (2), 026019.10.1088/1748-3190/abb86dCrossRefGoogle Scholar
Wu, T.Y.-T. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321–344.10.1017/S0022112061000949CrossRefGoogle Scholar
Wu, X., Zhang, X., Tian, X., Li, X. & Lu, W. 2020 A review on fluid dynamics of flapping foils. Ocean Engng 195, 106712.10.1016/j.oceaneng.2019.106712CrossRefGoogle Scholar
Wynn, R.B. 2014 Autonomous underwater vehicles (AUVs): their past, present and future contributions to the advancement of marine geoscience. Mar. Geol. 352, 451–468.10.1016/j.margeo.2014.03.012CrossRefGoogle Scholar
Yoerger, D.R. 2021 A hybrid underwater robot for multidisciplinary investigation of the ocean twilight zone. Sci. Robot. 6 (55), eabe1901.10.1126/scirobotics.abe1901CrossRefGoogle ScholarPubMed
Zhong, Q., Zhu, J., Fish, F.E., Kerr, S.J., Downs, A.M., Bart-Smith, H. & Quinn, D.B. 2021 Tunable stiffness enables fast and efficient swimming in fish-like robots. Sci. Robot. 6 (57), eabe4088.10.1126/scirobotics.abe4088CrossRefGoogle ScholarPubMed
Zhu, B., Wang, F., Xu, W. & Xiao, Q. 2023 Mode transition mechanisms in semi-passive flapping-wing propulsion or energy extraction. Phys. Fluids 35 (12), 123604.10.1063/5.0177197CrossRefGoogle Scholar
Zhu, J., White, C., Wainwright, D.K., Di Santo, V., Lauder, G.V. & Bart-Smith, H. 2019 Tuna robotics: a high-frequency experimental platform exploring the performance space of swimming fishes. Sci. Robot. 4 (34), eaax4615.10.1126/scirobotics.aax4615CrossRefGoogle ScholarPubMed