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In this paper, we report the results from the analysis of a numerical model used for the design of a magnetic linear actuator with applications to fi n-based locomotion. Most of the current robotic fi sh generate bending motion using rotary motors that implies at least one mechanical conversion of the motion. We seek a solution that directly bends the fi n and, at the same time, is able to exploit the magneto-mechanical properties of the fi n material. This strong fi n–actuator coupling blends the actuator and the body of the robot, allowing cross optimization of the system’s elements.
We study a simplifi ed model of an elastic element, a spring–mass system representing a fl exible fi n, subjected to nonlinear forcing, emulating magnetic interaction. The dynamics of the system is studied under unforced and periodic forcing conditions. The analysis is focused on the limit cycles present in the system, which allows the periodic bending of the fi n and the generation of thrust. The frequency, maximum amplitude and center of the periodic orbits (offset of the bending) depend directly on the stiffness of the fi n and the intensity of the forcing; we use this dependency to sketch a simple parameter controller. Although the model is strongly simplifi ed, it provides means to estimate fi rst values of the parameters for this kind of actuator and it is useful to evaluate the feasibility of minimal actuation control of such systems.
Fin-based locomotion, fl exible fi n, Magneto-Mechanical Actuators, Robotic fish.
[1] Cheng, J.Y., Pedley, T.J. & Altringham, J.D., A continuous dynamic beam model for swimming fi sh. Philosophical Transactions: Biological Sciences, 353(1371), pp. 981–997, 1998. doi: http:// dx.doi.org/10.1098/rstb.1998.0262
[2] McMillen, T. & Holmes, P., An elastic rod model for anguilliform swimming. Journal of Mathematical Biology, 53(5), pp. 843–886, 2006. doi: http://dx.doi.org/10.1007/s00285-006-0036-8
[3] Eldredge, J.D., A reconciliation of viscous and inviscid approaches to computing locomotion of deforming bodies. Experimental Mechanics, 50(9), pp. 1349–1353, 2009. doi: http://dx.doi. org/10.1007/s11340-009-9275-0
[4] Kanso, E. & Newton, P.K., Passive locomotion via normal-mode coupling ina submerged spring-mass system. Journal of Fluid Mechanics, 641, p. 205, 2009. doi: http://dx.doi.org/
10.1017/S0022112009992357
[5] Kanso, E., Swimming in an inviscid fl uid. Theoretical and Computational Fluid Dynamics, 24(1), pp. 201–207, 2009.
[6] Shukla, R.K. & Eldredge, J.D., An inviscid model for vortex shedding from a deforming body. Theoretical and Computational Fluid Dynamics, 21(5), p. 343, 2007. doi: http://dx.doi. org/10.1007/s00162-007-0053-2
[7] Eldredge, J.D. & Pisani, D., Passive locomotion of a simple articulated fi shlike system in the wake of an obstacle. Journal of Fluid Mechanics, 607, pp. 279–288, 2008. doi: http://dx.doi. org/10.1017/S0022112008002218
[8] Alben, S., Passive and active bodies in vortex-street wakes. Journal of Fluid Mechanics, 642(1), pp. 95–125, 2010. doi: http://dx.doi.org/10.1017/S0022112009991741
[9] Ahlborn, B., Chapman, S., Stafford, R. & Harper, R., Experimental simulation of the thrust phases of fast-start swimming of fi sh. The Journal of Experimental Biology, 200(17), pp. 2301–2312, 1997.
[10] Lauder, G.V., Anderson, E.J., Tangorra, J. & Madden, P.G.A., Fish biorobotics: kinematics and hydrodynamics of self-propulsion. The Journal of Experimental Biology, 210(Pt 16), pp. 2767–2780, 2007. doi: http://dx.doi.org/10.1242/jeb.000265
[11] Epps, B.P., Alvarado, P.V.Y., Youcef-Toumi, K. & Techet, A.H., Swimming performance of a biomimetic compliant fi sh-like robot. Experiments in Fluids, 47(6), p. 927, 2009. doi: http://dx. doi.org/10.1007/s00348-009-0684-8
[12] Deng, X. & Avadhanula, S., Biomimetic micro underwater vehicle with oscillating fi n propulsion: system design and force measurement. Proceedings 2005 IEEE International Conference on Robotics and Automation, pp. 3312–3317, 2005. doi: http://dx.doi.org/10.1109/ ROBOT.2005.1570621
[13] Alben, S., On the swimming of a fl exible body in a vortex street. Journal of Fluid Mechanics, 635(1), pp. 27–45, 2009. doi: http://dx.doi.org/10.1017/S0022112009990619
[14] Richardson, W., Greene, C., Malme, C. & Thomson, D., Marine Mammals and Noise, Academic Press: London and San Diego, 1995.
[15] Allen, J.J. & Smits, A.J., Energy harvesting eel. Journal of Fluids and Structures, 15(3–4), pp. 629–640, 2001. doi: http://dx.doi.org/10.1006/jfl s.2000.0355
[16] Beal, D.N., Hover, F.S., Triantafyllou, M.S., Liao, J.C. & Lauder, G.V., Passive propulsion in vortex wakes. Journal of Fluid Mechanics, 549(1), p. 385, 2006. doi: http://dx.doi.org/10.1017/ S0022112005007925
[17] Liao, J.C., A review of fi sh swimming mechanics and behaviour in altered fl ows. Philosophical Transactions of the Royal Society B, 362(1487), p. 1973, 2007. doi: http://dx.doi.org/10.1098/ rstb.2007.2082
[18] Vakakis, A., Gendelman, O., Bergman, L., McFarland, D., Kerschen, G. & Lee, Y., Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Solid Mechanics and Its Applications, vol. 156. Springer: Netherlands, 2009.
[19] Vokoun, D., Beleggia, M. & L. Heller, P.S., Magnetostatic interactions and forces between cylindrical permanent magnets. Journal of Magnetism and Magnetic Materials, 321(22), pp. 3758–3763, 2009. doi: http://dx.doi.org/10.1016/j.jmmm.2009.07.030
[20] Yadykin, Y., Tenetov, V. & Levin, D., The added mass of a fl exible plate oscillating in a fl uid. Journal of Fluids and Structures, 17(1), pp. 115–123, 2003. doi: http://dx.doi.org/10.1016/ S0889-9746(02)00100-7
[21] Righetti, L., Buchli, J. & Ijspeert, A., Dynamic hebbian learning in adaptive frequency oscillators. Physica D, 216(2), pp. 269–281, 2006. doi: http://dx.doi.org/10.1016/j.physd.2006.02.009