Hydrodynamics maneuver of a single helical flagellum swimming robot at low-Reynolds condition

  • Hassan SayyaadiEmail author
  • Shahnaz Bahmanyar
Research Paper


Helical swimming robots with a capable propulsion system at low-Reynolds numbers have been proposed for many applications. Although linear propulsion characteristics of swimming robots with a single helical flagellum have been extensively studied, the characteristics of maneuverability have not been completely investigated yet. This study presents a new method for the maneuverability of the helical swimming robot with a single helical flagellum. This mechanism is based on the change in the angle between the helical and body axes. This study shows that a change in the aforementioned angle can enable the swimming robot to have turning maneuvers in clockwise or counterclockwise directions. Moreover, the swimming robot will move in a straight line if the helical and body axes are parallel. To investigate this new method and predict the robot’s behavior at various inclination angles, a hydrodynamics model is used. To validate the hydrodynamics model, an experimental prototype of a macro-size swimming robot with specific inclination angles is fabricated. The results indicate that the helical swimming robot swims on circular trajectories through specific inclination angles between the helical flagellum and the body axis. Moreover, the radius of curvature decreases by increasing the inclination angle. Results of the validated hydrodynamics model indicate that the turning velocity has approximately a constant value at different inclination angles depending on the rotational frequency and geometrical parameters of the swimming robot. Finally, the effects of geometrical parameters of the body and the helical flagellum on the radius of curvature and turning velocity are investigated through the proposed hydrodynamics model. The verified results indicate that the hydrodynamics model provides a viable alternative model to predict the behavior of a helical swimming robot at various inclination angles within a range of design variables. This new method can be introduced as a mechanism for maneuverability of the helical swimming robots with a single helical flagellum and will be able to control the parameters in this type of swimmers for the implementation of predefined missions.


Helical swimmer robot Single helical flagellum Hydrodynamics maneuver Flagellum inclination angle 



Cylindrical body diameter (mm)


Cylindrical body length (mm)


Helical tail diameter (mm)


Helical wave amplitude (mm)


Helical tail’s length (mm)


Overall length of swimmer robot (body length + helical tail) (mm)


Pitch angle (°)


Inclination angle (°)


Helical wave length (mm)


Number of wavelengths (−)


Total weight (g)


DC-motor diameter (mm)


DC-motor length (mm)


Voltage of motor (V)


Volume of battery (m3)


Voltage of battery (V)


Density of test fluid (Kg/m3)


Kinematic viscosity (cSt)


Spinal propulsive frequency (Hz)


The center of mass \(G=\left({x}_{g},{y}_{g},{z}_{g}\right)\) (mm)


The center of buoyancy \(B=\left({x}_{B},{y}_{B},{z}_{B}\right)\)(mm)

\({\overrightarrow{F}}_{\rm helix}\)

Propulsive force in x-direction

\({\overrightarrow{M}}_{\rm helix}\)

Torque resulting from fluid reaction on the helical tail

\({\overrightarrow{F}}_{\rm body}, {\overrightarrow{M}}_{\rm body}\)

Viscous drag and torque acting on the body

\({\overrightarrow{F}}_{\rm e}, {\overrightarrow{M}}_{\rm e}\)

External forces and torques that affect the swimmer

\({df}_{n}, {df}_{t}\)

Hydrodynamic forces acting on a cylindrical element of local length

\({ \xi }_{n}, { \xi }_{t}\)

Local drag coefficient for motion normal and tangential to local length

\({ \upsilon }_{n}, { \upsilon }_{t}\)

Components of local normal and tangential to local length (mm/s)


The resistive matrix for the body


The resistive matrix for the helical flagella


Angular velocity of the swimmer robot in inertial coordinates \({\Omega }=\left(\dot{{\theta }},\dot{{\phi }},\dot{{\Psi }}\right)\) (rad/s)


Radius of curvature (mm)


Planer velocity of swimmer robot \((\text{mm/s})\)


Velocity of the swimmer robot in body-fixed coordinates \(v=\left({v}_{x},{v}_{y},{v}_{z}\right)\) \((\text{mm/s})\)


Velocity of the swimmer robot in inertial coordinates \(V=\left({V}_{x},{V}_{y},{V}_{z}\right)\) \((\text{mm/s})\)


Angular velocity of the swimmer robot in body-fixed coordinates \(\varOmega =\left({\varOmega }_{x},{\varOmega }_{y},{\varOmega }_{z}\right)\)(\(\text{rad/s}\))



  1. Batchelor GK (1970), Slender-body theory for particles of arbitrary cross-section in Stokes flow. J Fluid Mech 44(3):419MathSciNetCrossRefGoogle Scholar
  2. Behkam B, Sitti M (2006) Design methodology for biomimetic propulsion of miniatur swimming robot. Trans ASME J Dyn Sys Meas Control 128:36–43CrossRefGoogle Scholar
  3. Berg H (2003) The rotary motor of bacterial flagella. Ann Rev Biochem 72:19–54CrossRefGoogle Scholar
  4. Brennen C, Winet H (1977) Fluid mechanics of propulsion by cilia and flagella. Annu Rev Fluid Mech 9:339–398CrossRefGoogle Scholar
  5. Chen B, Jiang S, Liu Y, Yang P, Chen S (2010) Research on the kinematic properties of a sperm-like swimming micro robot. J Bionic Eng 7:S123–S129CrossRefGoogle Scholar
  6. Chwang AT, Wu TY (1971) A note on the helical movement of micro-organisms. Proc R Soc Lond B 178:327–346CrossRefGoogle Scholar
  7. Darnton NC, Turner L, Rojevsky S, Berg HC (2007) On torque and tumbling in swimming Escherichia coli. J Bacteriol 189:1756–1764CrossRefGoogle Scholar
  8. Edd J, Payen S, Stoller M, Rubinsky B, Sitti M (2003) Biomimetic propulsion mechanism for a swimming surgical micro-robot. In: Proc.IEEE/RSJ Int. Conf. Intell. Rob. Syst., Las Vegas, NV, USA, pp 2583–2588Google Scholar
  9. Elgeti J, Winkler RG, Gompper G (2015) Physics of microswimmers-single particle motion and collective behavior: review. Rep Prog Phys 78:056601MathSciNetCrossRefGoogle Scholar
  10. Feng J, Cho SK (2014) Mini and micro propulsion for medical swimmers. Micromachines 5:97–113. CrossRefGoogle Scholar
  11. Garcia J, Torre DL, Bloomfield VA (1977) Hydrodynamic theory of swimming of flagellated microorganism. Biophys J 20:49CrossRefGoogle Scholar
  12. Gray J, Hancock GJ (1955) The propulsion of sea-urchin spermatozoa. J Exp Biol 32:802Google Scholar
  13. Ha N, Goo N, Yoon H (2011) Development of a propulsion system for a biomimetic thruster. Chinese Sci Bull 56:432–438CrossRefGoogle Scholar
  14. Johnson RE, Brokaw CJ (1979) Flagellar hydrodynamics a comparison between resistive force theory and Slender body theory. Biophys Soc 25:113–127Google Scholar
  15. Keller J, Rubinow S (1976) Swimming of flagellated microorganisms. Biophys J 16:151CrossRefGoogle Scholar
  16. Lagua E (2016) Bacterial hydrodynamics. Annu Rev Fluid Mech 48:105–130MathSciNetCrossRefGoogle Scholar
  17. Lighthill J, Lighthill M (1975) Mathematical biofluid dynamics. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  18. Liou W, Yang Y (2015) Numerical study of low-Reynolds number flow over rotating rigid helix: an investigation of the unsteady hydrodynamic force. Fluid Dyn Res 47:045506 (IOP publication)MathSciNetCrossRefGoogle Scholar
  19. McCarter L, Hilmen M, Silverman M (1988) Flagellar dynamometer controls swarmer cell differentiation of V. parahaemolyticus. Cell 54:345–351CrossRefGoogle Scholar
  20. Nelson BJ, Kaliakatsos I, Abbott JJ (2010) Micro robots for minimally invasive medicine. Annu Rev Biomed Eng 12:55–85CrossRefGoogle Scholar
  21. Nourmohammadi H, Keighobadi J, Bahrami M (2016) Design, dynamic modelling and control of a bio-inspired helical swimming microrobot with three-dimensional manoeuvring. Trans Inst Meas Control (SAGE) 39(Issue 7):1036–1046Google Scholar
  22. Pak Sh, Lauga E (2014) Theoretical models in low-Reynolds-number locomotion. In: Duprat C, Stone HA (eds) Low-Reynolds-number flows: fluid-structure interactions. Soft Matter Series. Royal Society of ChemistryGoogle Scholar
  23. Peyer KE, Mahoney AW, Zhang LJ, Abbott BJ, Nelson (2012) Bacteria-inspired microrobots. Microbiorobotics. (Elsevier Inc)CrossRefGoogle Scholar
  24. Purcell EM (1997) The efficiency of propulsion by a rotating flagellum. Natl Acad Sci USA (PNAS) 94:11307–11311CrossRefGoogle Scholar
  25. Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Philosoph Trans R Soc Lond 174:935–982CrossRefGoogle Scholar
  26. Sitti M (2007) Microscale and nanoscale robotics systems [grand challenges of robotics. IEEE Robot Autom Mag 14(1):53–60CrossRefGoogle Scholar
  27. Tabak AF, Yesilyurt S (2013) Improved Kinematic models for two-link helical micro/nano-swimmers. IEEE Transact Robot 30:14–25CrossRefGoogle Scholar
  28. Taute KM, Gude S, Tans SJ, Shimizu TS (2015) High-throughput 3D tracking of bacteria on a standard phase contrast microscope. Nature Commun 6:8776CrossRefGoogle Scholar
  29. Temel FZ, Yesilyurt S (2013) Simulation-based analysis microrobots swimming at the center and near the wall of circular minichannels. Microfluid Nanofluid 14(1–2):287–298CrossRefGoogle Scholar
  30. Tottori S, Zhang L, Qiu F, Krawczyk K, Obregón Al, Nelson BJ (2012) Magnetic helical micromachines: fabrication, controlled swimming, and cargo transport. Adv Mater 24:811–816CrossRefGoogle Scholar
  31. Xie L, Altindal T, Chattopadhyay S, Wu X (2011) Bacterial flagellum as a propeller and as a rudder for efficient chemo taxis. PNAS 108(6):2246–2251CrossRefGoogle Scholar
  32. Xu T, Hwang G, Andreff N, R´egnier S (2015) Influence of geometry on swimming performance of helical Swimmers using DoE, Springer. J Microbiorobotic 31(1):117–127Google Scholar
  33. Ye Z, R´egnier St, Sitti M (2013) Rotating magnetic miniature swimming robots with multiple flexible flagella. IEEE Trans Rovotics.[22]
  34. Zhang L, Peyer K, Nelson BJ (2010) Artificial bacterial flagella for micromanipulation. Lab Chip 10(17):2203–2215CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Center of Excellence in Hydrodynamics and Dynamics of Marine VehiclesSharif University of TechnologyTehranIran

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