Skip to main content
SpringerLink
Log in

Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We consider the controlled motion in an ideal incompressible fluid of a rigid body with moving internal masses and an internal rotor in the presence of circulation of the fluid velocity around the body. The controllability of motion (according to the Rashevskii–Chow theorem) is proved for various combinations of control elements. In the case of zero circulation, we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion. In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating the drift) at the end point of the trajectory. We show that the drift can be compensated for if the body is inside a circular domain whose size is defined by the geometry of the body and the value of circulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. D. Ageev, L. V. Kiselev, Yu. V. Matvienko, et al., Autonomous Underwater Robots: Systems and Technology, Ed. by M. D. Ageev (Nauka, Moscow, 2005) [in Russian].

  2. A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint (Springer, Berlin, 2004).

    Book  MATH  Google Scholar 

  3. R. McN. Alexander, “The history of fish mechanics,” in Fish Biomechanics, Ed. by P. W. Webb and D. Weihs (Praeger, New York, 1983), pp. 1–35.

    Google Scholar 

  4. G. Antonelli, Underwater Robots (Springer, Cham, 2014).

    Book  Google Scholar 

  5. H. C. Berg, Random Walks in Biology (Princeton Univ. Press, Princeton, NJ, 1993).

    Google Scholar 

  6. P. Bhatta and N. E. Leonard, “Stabilization and coordination of underwater gliders,” in Proc. 41st IEEE Conf. on Decision and Control (IEEE, Piscataway, NJ, 2002), Vol. 2, pp. 2081–2086.

    Google Scholar 

  7. R. W. Blake, Fish Locomotion (Cambridge Univ. Press, Cambridge, 1983).

    Google Scholar 

  8. N. N. Bolotnik, T. Yu. Figurina, and F. L. Chernous’ko, “Optimal control of the rectilinear motion of a two-body system in a resistive medium,” Prikl. Mat. Mekh. 76(1), 3–22 (2012) [J. Appl. Math. Mech. 76, 1–14 (2012)].

    MathSciNet  MATH  Google Scholar 

  9. B. Bonnard, “Contrôlabilité des systèmes non linéaires,” C. R. Acad. Sci. Paris, Sér. 1, 292, 535–537 (1981).

    MathSciNet  MATH  Google Scholar 

  10. B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory (Springer, Paris, 2003), Math. Appl. 40.

    MATH  Google Scholar 

  11. A. V. Borisov, A. Yu. Jalnine, S. P. Kuznetsov, I. R. Sataev, and Ju. V. Sedova, “Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback,” Regul. Chaotic Dyn. 17(6), 512–532 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “New effects in dynamics of rattlebacks,” Dokl. Akad. Nauk 408(2), 192–195 (2006) [Dokl. Phys. 51(5), 272–275 (2006)].

    MathSciNet  MATH  Google Scholar 

  13. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “How to control Chaplygin’s sphere using rotors,” Nelinein. Din. 8(2), 289–307 (2012) [Regul. Chaotic Dyn. 17(3–4), 258–272 (2012)].

    Article  MATH  Google Scholar 

  14. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “How to control the Chaplygin ball using rotors. II,” Nelinein. Din. 9(1), 59–76 (2013) [Regul. Chaotic Dyn. 18(1–2), 144–158 (2013)].

    Article  MathSciNet  MATH  Google Scholar 

  15. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “The problem of drift and recurrence for the rolling Chaplygin ball,” Nelinein. Din. 9(4), 721–754 (2013) [Regul. Chaotic Dyn. 18(6), 832–859 (2013)].

    Article  MATH  Google Scholar 

  16. A. V. Borisov, V. V. Kozlov, and I. S. Mamaev, “Asymptotic stability and associated problems of dynamics of falling rigid body,” Regul. Chaotic Dyn. 12(5), 531–565 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. V. Borisov and I. S. Mamaev, Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos (Inst. Komp’yut. Issled., Moscow, 2005) [in Russian].

    MATH  Google Scholar 

  18. A. V. Borisov and I. S. Mamaev, “On the motion of a heavy rigid body in an ideal fluid with circulation,” Chaos 16(1), 013118 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. V. Borisov, I. S. Mamaev, and S. M. Ramodanov, “Motion of a circular cylinder and n point vortices in a perfect fluid,” Regul. Chaotic Dyn. 8(4), 449–462 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. V. Borisov, I. S. Mamaev, and S.M. Ramodanov, “Dynamic interaction of point vortices and a two-dimensional cylinder,” J. Math. Phys. 48(6), 065403 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Canudas de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Autom. Control 40(3), 419–425 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. A. Chaplygin, On the Effect of a Plane-Parallel Air Flow on a Cylindrical Wing Moving in It (Moscow, 1926), Tr. Tsentr. Aerohydrodin. Inst. 19; reprint. in Collected Works (Gostekhizdat, Moscow, 1948), Vol. 2, pp. 300–382 [in Russian].

    Google Scholar 

  23. S. A. Chaplygin, “On the motion of heavy bodies in an incompressible fluid,” in Collected Works (Gostekhizdat, Moscow, 1948), Vol. 1, pp. 312–336 [in Russian].

    Google Scholar 

  24. F. L. Chernous’ko, “The optimal periodic motions of a two-mass system in a resistant medium,” Prikl. Mat. Mekh. 72(2), 202–215 (2008) [J. Appl. Math. Mech. 72, 116–125 (2008)].

    MathSciNet  MATH  Google Scholar 

  25. S. Childress, S. E. Spagnolie, and T. Tokieda, “A bug on a raft: Recoil locomotion in a viscous fluid,” J. Fluid Mech. 669, 527–556 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  26. W.-S. Chu et al., “Review of biomimetic underwater robots using smart actuators,” Int. J. Precis. Eng. Manuf. 13(7), 1281–1292 (2012).

    Article  Google Scholar 

  27. M. Chyba, N. E. Leonard, and E. D. Sontag, “Optimality for underwater vehicles,” in Proc. 40th IEEE Conf. on Decision and Control (IEEE, Piscataway, NJ, 2001), Vol. 5, pp. 4204–4209.

    Article  Google Scholar 

  28. J. Cochran, E. Kanso, S. D. Kelly, H. Xiong, and M. Krstic, “Source seeking for two nonholonomic models of fish locomotion,” IEEE Trans. Rob. 25(5), 1166–1176 (2009).

    Article  Google Scholar 

  29. J. E. Colgate and K. M. Lynch, “Mechanics and control of swimming: A review,” IEEE J. Oceanic Eng. 29(3), 660–673 (2004).

    Article  Google Scholar 

  30. P. E. Crouch, “Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models,” IEEE Trans. Autom. Control 29(4), 321–331 (1984).

    Article  MATH  Google Scholar 

  31. K. D. Do, Z. P. Jiang, J. Pan, and H. Nijmeijer, “A global output-feedback controller for stabilization and tracking of underactuated ODIN: A spherical underwater vehicle,” Automatica 40(1), 117–124 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  32. K. M. Ehlers and J. Koiller, “Micro-swimming without flagella: Propulsion by internal structures,” Regul. Chaotic Dyn. 16(6), 623–652 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Föppl, “Wirbelbewegung hinter einem Kreiszylinder,” Münch. Ber. 1913, 1–17 (1913).

    MATH  Google Scholar 

  34. J. Gray, “Studies in animal locomotion. VI: The propulsive powers of the dolphin,” J. Exp. Biol. 13(2), 192–199 (1936).

    Google Scholar 

  35. T. H. Havelock, “The stability of motion of rectilinear vortices in ring formation,” Philos. Mag., Ser. 7, 11 (70), Suppl., 617–633 (1931).

    Article  MATH  Google Scholar 

  36. P. Jagadeesh, K. Murali, and V. G. Idichandy, “Experimental investigation of hydrodynamic force coefficients over AUV hull form,” Ocean Eng. 36(1), 113–118 (2009).

    Article  Google Scholar 

  37. M. A. Jones and M. J. Shelley, “Falling cards,” J. Fluid Mech. 540, 393–425 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  38. V. Jurdjevic, Geometric Control Theory (Cambridge Univ. Press, Cambridge, 1997).

    MATH  Google Scholar 

  39. E. Kanso, J. E. Marsden, C. W. Rowley, and J. B. Melli-Huber, “Locomotion of articulated bodies in a perfect fluid,” J. Nonlinear Sci. 15(4), 255–289 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  40. Yu. L. Karavaev and A. A. Kilin, “The dynamics and control of a spherical robot with an internal omniwheel platform,” Nelinein. Din. 11(1), 187–204 (2015) [Regul. Chaotic Dyn. 20(2), 134–152 (2015)].

    Article  MATH  Google Scholar 

  41. A. O. Kazakov, “Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane,” Regul. Chaotic Dyn. 18(5), 508–520 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. A. Kilin, E. N. Pivovarova, and T. B. Ivanova, “Spherical robot of combined type: Dynamics and control,” Regul. Chaotic Dyn. 20(6), 716–728 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  43. A. A. Kilin and E. V. Vetchanin, “The control of the motion through an ideal fluid of a rigid body by means of two moving masses,” Nelinein. Din. 11(4), 633–645 (2015).

    Article  MATH  Google Scholar 

  44. L. V. Kiselev and A. V. Medvedev, “Comparative analysis and optimization of the dynamical properties of autonomous underwater robots of various projects and configurations,” Podvodn. Issled. Robototekh., No. 1, 24–35 (2012).

    Google Scholar 

  45. G. Kirchhoff, Vorlesungen über mathematische Physik: Mechanik (B.G. Teubner, Leipzig, 1877).

    MATH  Google Scholar 

  46. J. Koiller, K. Ehlers, and R. Montgomery, “Problems and progress in microswimming,” J. Nonlinear Sci. 6(6), 507–541 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  47. V. V. Kozlov and D. A. Onishchenko, “Motion of a body with rigid shell and variable mass geometry in an unbounded perfect fluid,” in Problems of Mechanics: Collection of Papers Dedicated to the 90th Birthday of A.Yu. Ishlinskii (Fizmatlit, Moscow, 2003), pp. 465–476 [in Russian].

    Google Scholar 

  48. V. V. Kozlov and S. M. Ramodanov, “The motion of a variable body in an ideal fluid,” Prikl. Mat. Mekh. 65(4), 592–601 (2001) [J. Appl. Math. Mech. 65, 579–587 (2001)].

    MathSciNet  MATH  Google Scholar 

  49. V. V. Kozlov and S. M. Ramodanov, “On the motion of a body with a rigid shell and variable mass geometry in a perfect fluid,” Dokl. Akad. Nauk 382(4), 478–481 (2002) [Dokl. Phys. 47(2), 132–135 (2002)].

    Google Scholar 

  50. R. P. Kumar, A. Dasgupta, and C. S. Kumar, “Real-time optimal motion planning for autonomous underwater vehicles,” Ocean Eng. 32(11–12), 1431–1447 (2005).

    Article  Google Scholar 

  51. H. Lamb, Hydrodynamics, 6th ed. (Dover Publ., New York, 1945).

    MATH  Google Scholar 

  52. N. E. Leonard, “Stability of a bottom-heavy underwater vehicle,” Automatica 33(3), 331–346 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  53. N. E. Leonard and J. E. Marsden, “Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry,” Physica D 105(1–3), 130–162 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  54. A. M. Lyapunov, “On constant helical motions of a rigid body in a fluid,” Soobshch. Kharkov. Mat. Obshch., Ser. 2, 1(1), 7–60 (1888).

    Google Scholar 

  55. S. Michelin and S. G. Llewelly Smith, “An unsteady point vortex method for coupled fluid–solid problems,” Theor. Comput. Fluid Dyn. 23(2), 127–153 (2009).

    Article  MATH  Google Scholar 

  56. R. M. Murray, J. W. Burdick, S. D. Kelly, and J. Radford, “Trajectory generation for mechanical systems with application to robotic locomotion,” in Robotics: The Algorithmic Perspective: Proc. 3rd Workshop on the Algorithmic Foundations of Robotics (A.K. Peters, Natick, MA, 1998), pp. 81–90.

    Google Scholar 

  57. L. Paull, S. Saeedi, M. Seto, and H. Li, “AUV navigation and localization: A review,” IEEE J. Oceanic Eng. 39(1), 131–149 (2014).

    Article  Google Scholar 

  58. E. M. Purcell, “Life at low Reynolds number,” Am. J. Phys. 45(1), 3–11 (1977).

    Article  MathSciNet  Google Scholar 

  59. S. M. Ramodanov and V. A. Tenenev, “The motion of a two-dimensional body, controlled by two moving internal masses, in an ideal fluid,” Prikl. Mat. Mekh. 79(4), 463–475 (2015) [J. Appl. Math. Mech. 79, 325–333 (2015)].

    MathSciNet  Google Scholar 

  60. S. M. Ramodanov, V. A. Tenenev, and D. V. Treschev, “Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid,” Nelinein. Din. 8(4), 799–813 (2012) [Regul. Chaotic Dyn. 17(6), 547–558 (2012)].

    Article  MATH  Google Scholar 

  61. P. K. Rashevskii, “Connectibility of any two points of a totally nonholonomic space by an admissible curve,” Uch. Zap. Mosk. Gos. Pedagog. Inst., Ser. Fiz.-Mat. 3(2), 83–94 (1938).

    Google Scholar 

  62. P. G. Saffman, “The self-propulsion of a deformable body in a perfect fluid,” J. Fluid Mech. 28(2), 385–389 (1967).

    Article  MATH  Google Scholar 

  63. P. G. Saffman, Vortex Dynamics (Cambridge Univ. Press, Cambridge, 1992).

    MATH  Google Scholar 

  64. A. V. Sakharov, “Rotation of the body with movable internal masses around the center of mass on a rough plane,” Regul. Chaotic Dyn. 20(4), 428–440 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  65. M. Sfakiotakis, D. M. Lane, and J. B. C. Davies, “Review of fish swimming modes for aquatic locomotion,” IEEE J. Oceanic Eng. 24(2), 237–252 (1999).

    Article  Google Scholar 

  66. B. N. Shashikanth, “Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and N point vortices: The case of arbitrary smooth cylinder shapes,” Regul. Chaotic Dyn. 10(1), 1–14 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  67. B. N. Shashikanth, A. Sheshmani, S. D. Kelly, and J. E. Marsden, “Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: The case of arbitrary smooth body shape,” Theor. Comput. Fluid Dyn. 22(1), 37–64 (2008).

    Article  MATH  Google Scholar 

  68. J. A. Sparenberg, “Survey of the mathematical theory of fish locomotion,” J. Eng. Math. 44(4), 395–448 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  69. V. A. Steklov, “Motion of a heavy rigid body in a fluid,” Soobshch. Kharkov. Mat. Obshch., Ser. 2, 2(1–2), 209–235 (1889).

    Google Scholar 

  70. V. A. Steklov, “On some possible motions of a rigid body in a fluid,” Tr. Otd. Fiz. Nauk Obshch. Lyubitelei Estestvoznaniya 7(2), 10–21 (1895).

    Google Scholar 

  71. Swimming and Flying in Nature, Ed. by T. Y.-T. Wu, C. J. Brokaw, and C. Brennen (Plenum, New York, 1975), Vol.1.

  72. P. Tallapragada and S. D. Kelly, “Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid,” Regul. Chaotic Dyn. 18(1–2), 21–32 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  73. P. Tallapragada and S. D. Kelly, “Self-propulsion of free solid bodies with internal rotors via localized singular vortex shedding in planar ideal fluids,” Eur. Phys. J. Spec. Top. 224(17), 3185–3197 (2015).

    Article  Google Scholar 

  74. V. A. Tenenev, E. V. Vetchanin, and L. F. Ilaletdinov, “Chaotic dynamics in the problem of the fall of a screw-shaped body in a fluid,” Nelinein. Din. 12(1), 99–120 (2016).

    Article  MATH  Google Scholar 

  75. M. S. Triantafyllou, A. H. Techet, and F. S. Hover, “Review of experimental work in biomimetic foils,” IEEE J. Oceanic Eng. 29(3), 585–594 (2004).

    Article  Google Scholar 

  76. M. S. Triantafyllou, G. S. Triantafyllou, and D. K. P. Yue, “Hydrodynamics of fishlike swimming,” Annu. Rev. Fluid Mech. 32, 33–53 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Vankerschaver, E. Kanso, and J. Marsden, “The geometry and dynamics of interacting rigid bodies and point vortices,” J. Geom. Mech. 1(2), 223–266 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  78. J. Vankerschaver, E. Kanso, and J. E. Marsden, “The dynamics of a rigid body in potential flow with circulation,” Regul. Chaotic Dyn. 15(4–5), 606–629 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  79. E. V. Vetchanin and A. A. Kilin, “Free and controlled motion of a body with a moving internal mass through a fluid in the presence of circulation around the body,” Dokl. Akad. Nauk 466(3), 293–297 (2016) [Dokl. Phys. 61(1), 32–36 (2016)].

    MathSciNet  Google Scholar 

  80. E. V. Vetchanin and A. A. Kilin, “Control of body motion in an ideal fluid using the internal mass and the rotor in the presence of circulation around the body,” J. Dyn. Control Syst., doi: 10.1007/s10883-016-9345-4 (2016).

    Google Scholar 

  81. E. V. Vetchanin, I. S. Mamaev, and V. A. Tenenev, “The self-propulsion of a body with moving internal masses in a viscous fluid,” Nelinein. Din. 8(4), 815–836 (2012) [Regul. Chaotic Dyn. 18(1–2), 100–117 (2013)].

    Article  MATH  Google Scholar 

  82. C. A. Woolsey and N. E. Leonard, “Stabilizing underwater vehicle motion using internal rotors,” Automatica 38(12), 2053–2062 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  83. T. Y. Wu, “Fish swimming and bird/insect flight,” Annu. Rev. Fluid Mech. 43, 25–58 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  84. J. Yu, M. Tan, S. Wang, and E. Chen, “Development of a biomimetic robotic fish and its control algorithm,” IEEE Trans. Syst. Man Cybern. B 34(4), 1798–1810 (2004).

    Article  Google Scholar 

  85. C. Zhou and K. H. Low, “Design and locomotion control of a biomimetic underwater vehicle with fin propulsion,” IEEE/ASME Trans. Mechatron. 17(1), 25–35 (2012).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069, Russia

    E. V. Vetchanin

  2. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    E. V. Vetchanin & A. A. Kilin

Authors
  1. E. V. Vetchanin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Kilin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. V. Vetchanin.

Additional information

Original Russian Text © E.V. Vetchanin, A.A. Kilin, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 321–351.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vetchanin, E.V., Kilin, A.A. Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid. Proc. Steklov Inst. Math. 295, 302–332 (2016). https://doi.org/10.1134/S0081543816080186

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543816080186

Search

Navigation