Abstract
Micro-robots that can propel themselves in a low Reynolds number fluid flow by converting their rotational motion into translation have begun attracting much attention due to their ease of fabrication. The dynamics and controllability of the motion of such microswimmers are investigated in this paper. The microswimmers under consideration here are spinning spheres (or rotors) whose dynamics are approximated by rotlets, a singularity solution of the Stokes equations. While singularities of Stokes flows are commonly used as theoretical models for microswimmers and micro-robots, rotlet models of microswimmers have received less attention. While a rotlet alone cannot generate translation, a pair of rotlets can interact and execute net motion. Taking the control inputs to be the strengths of the micro rotors, the positions of a pair of rotors are not controllable in an unbounded planar fluid domain. However, in a bounded domain, which is often the case of practical interest, we show that the positions of the micro rotors are controllable. This is enabled by the interaction of the rotors with the boundaries of the domain. We show how control inputs can be constructed based on combinations of Lie brackets to move the rotors from one point to another in the domain. Another contribution of this paper is the creation of a framework for path planning and control of the motion of Stokes singularities that model the dynamics of microswimmers. This can be extended to microswimmers with other shapes moving in confined fluid domains with complex boundaries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batchelor, G.K.: The stress system in a suspension of force-free particles. J. Fluid Mech. 41(3), 545570 (1970)
Blake, J.R., Chwang, A.T.: Fundamental singularities of viscous flow. J. Engi. Math. 8(1), 23–29 (1974)
Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, Berlin Heidelberg (2003)
Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems. Springer, Berlin (2004)
Cheang, U.K., Meshkati, F., Kim, D., Kim, M.J., Fu, H.C.: Minimal geometric requirements for micropropulsion via magnetic rotation. Phys. Rev. E 90(3), 033007 (2014)
Chen, X.Z., Hoop, M., Mushtaq, F., Siringil, E., Hu, C., Nelson, B.J., Pané, S.: Recent developments in magnetically driven micro- and nanorobots. Appl. Mater. Today 9, 37–48 (2017)
Chwang, A.T., Wu, T.Y.: Hydromechanics of low-reynolds-number flow. Part 2. Singularity method for stokes flows. J. Fluid Mech. 67(4), 787815 (1975)
Ding, Y., Qiu, F., Solvas, X.C., Chiu, F.W.Y., Nelson, B.J., de Mello, A.: Microfluidic-based droplet and cell manipulations using artificial bacterial flagella. Micromachines 7(2), 25 (2016)
Dreyfus, R., Baudry, J., Roper, M.L., Fermigier, M., Stone, H.A., Bibette, J.: Microscopic artificial swimmers. Nature 437, 862 (2005)
Fily, Y., Baskaran, A., Marchetti, M.C.: Cooperative self-propulsion of active and passive rotors. Soft Matter 8, 3002–3009 (2012)
Gao, W., Kagan, D., Clawson, C., Campuzano, S., Chuluun-Erdene, E., Shipton, E., Fullerton, E., Zhang, L., Lauga, E., Wang, J.: Cargo-towing fuel-free magnetic nanoswimmers for targeted drug delivery. Small. 8(3), 460–467 (2012)
Ghosh, A., Mandal, P., Karmakar, S., Ghosh, A.: Analytical theory and stability analysis of an elongated nanoscale object under external torque. Phys. Chem. Chem. Phys. 15, 10817 (2013)
Grzybowski, B.A., Stone, H.A., Whitesides, G.M.: Dynamics of self assembly of magnetized disks rotating at the liquid–air interface. PNAS 99(7), 4147–4151 (2002)
Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media (Mechanics of Fluids and Transport Processes). Springer, Berlin Heidelberg (1983)
Kadam, S., Joshi, K., Gupta, N., Katdare, P., Banavar, R.: Trajectory tracking using motion primitives for the purcell’s swimmer. In: 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS), pp. 3246–3251, Sept 2017
Kim, S., Karrila, S.J.: Microhydrodynamics: Principles and Selected Applications. Dover Publications, Mineola (2005)
Lauga, E., DiLuzio, W.R., Whitesides, G.M., Stone, H.A.: Swimming in circles: motion of bacteria near solid boundaries. Biophys. J. 90(2), 400–412 (2006)
Lauga, E., Powers, : The hydrodynamics of swimming microorganisms. Rep. Progress Phys. 72(9), 096601 (2009)
Leoni, M., Liverpool, T.B.: Dynamics and interactions of active rotors. EPL (Eur. Lett.) 92(6), 64004 (2010)
Lushi, E., Vlahovska, P.M.: Periodic and chaotic orbits of plane-confined micro-rotors in creeping flows. J. Nonlinear Sci. 25(5), 11111123 (2015)
Meleshko, V.V., Aref, H.: A blinking rotlet model for chaotic advection. Phys. Fluids 8, 3215 (1996)
Meshkati, F., Fu, H.: Modeling rigid magnetically rotated microswimmers: rotation axes, bistability, and controllability. Phys. Rev. E 90(6), 063006 (2014)
Murray, R., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)
Nelson, B.J., Kaliakatsos, I.K., Abbott, J.J.: Microrobots for minimally invasive medicine. Annu. Rev. Biomed. Eng. 12, 55–85 (2010)
Or, Y., Zhang, S., Murray, R.: Dynamics and stability of low-Reynolds-number swimming near a wall. SIAM J. Appl. Dyn. Syst. 10(3), 1013–1041 (2011)
Petit, T., Zhang, L., Peyer, K.E., Kratochvil, B.E., Nelson, B.J.: Selective trapping and manipulation of microscale objects using mobile microvortices. Nano Lett. 12(1), 156160 (2012)
Peyer, K.E., Zhang, L., Nelson, B.J.: Bio-inspired magnetic swimming microrobots for biomedical applications. Nanoscale 4(5), 1259–1272 (2013)
Purcell, E.M.: Life at low Reynolds number. Am. J. Phys. 45, 3–11 (1977)
Sastry, S.: Nonlinear Systems: Analysis, Stability, and Control. Springer, Berlin (1999)
Snezhko, A., Aranson, I.S.: Magnetic manipulation of self-assembled colloidal asters. Nat. Mater. 10, 698–703 (2011)
Spagnolie, S.E., Lauga, E.: Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105147 (2012)
Sudarsanam, S., Tallapragada, P.: Chaotic mixing using micro-rotors in a confined domain (2018) (submitted to Physics of Fluids)
Tierno, P., Golestanian, R., Pagonabarraga, I., Sagués, F.: Controlled swimming in confined fluids of magnetically actuated colloidal rotors. Phys. Rev. Lett. 101, 218304 (2008)
Zhang, L., Abbott, J.J., Dong, L., Kratochvil, B.E., Bell, D., Nelson, B.J.: Artificial bacterial flagella: fabrication and magnetic control. Appl. Phys. Lett. 94, 064107 (2009)
Zhang, L., Petit, T., Peyer, K.E., Nelson, B.J.: Targeted cargo delivery using a rotating nickel nanowire. Nanomedicine 8(7), 1074–80 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Buzhardt, J., Fedonyuk, V. & Tallapragada, P. Pairwise controllability and motion primitives for micro-rotors in a bounded Stokes flow. Int J Intell Robot Appl 2, 454–461 (2018). https://doi.org/10.1007/s41315-018-0075-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41315-018-0075-5