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Control of the motion of a triaxial ellipsoid in a fluid using rotors

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Abstract

The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.

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References

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

    MATH  MathSciNet  Google Scholar 

  2. V. V. Kozlov and S. M. Ramodanov, “On the motion of a body with a rigid shell and changing geometry of masses in an ideal fluid,” Dokl. Ross. Akad. Nauk 382 (4), 478–481 (2002) [Dokl. Math. 47 (2), 132–135 (2002)].

    Google Scholar 

  3. D. A. Onishchenko and V. V. Kozlov, “Motion in an ideal fluid of a body containing a moving lumped mass inside it,” Prikl. Mat. Mekh. 67 (4), 620–633 (2003) [J. Appl. Math. Mech. 67 (4), 553–564 (2003)].

    MATH  MathSciNet  Google Scholar 

  4. G. Kirchhoff, Vorlesungen über Mathematische Physik: Mechanik (B. G. Teubner, Leipzig, 1877).

    MATH  Google Scholar 

  5. 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,” Nelin. Dinam. 11 (4), 633–645 (2015).

    Article  MATH  Google Scholar 

  6. 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. Ross. Akad. Nauk 466 (3), 293–297 (2016) [Dokl. Phys. 61 (1), 32–36 (2016)].

    MATH  Google Scholar 

  7. 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. 23 (2), 435–458 (2017).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. T. B. Ivanova and E. N. Pivovarova, “Comments on the paper by A. V. Borisov, A. A. Kilin, and I. S. Mamaev “How to control the Chaplygin Ball Using Rotors. II”,” Regul. Chaotic Dyn. 19 (1), 140–143 (2014).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Morinaga, M. Svinin, and M. Yamamoto, “On the iterative steering a rolling robot actuated internal rotors,” in Springer Proc. Math. Stat., Vol. 121: Analysis, Modelling, Optimization, and Numerical Techniques (Springer, Cham, 2015), pp. 205–218.

    Google Scholar 

  12. P. K. Rashevskii, “On joining any two points of a nonholonomic space by an admissible line,” Uchen. Zap. Pedag. Inst. Liebkhecht Ser. Fiz.–Mat. 3 (2), 83–94 (1938).

    Google Scholar 

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

    Google Scholar 

  14. P. Holmes, J. Jenkins, and N. E. Leonard, “Dynamics of the Kirchhoff equations. I. Coincident centers of gravity and buoyancy,” Phys. D 118 (3-4), 311–342 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  15. J. L. G. Guirao and J. A. Vera, “Equilibria, stability and Hamiltonian Hopf bifurcation a gyrostat in an incompressible ideal fluid,” Phys. D 241 (19), 1648–1654 (2012).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Biggs and W. Holderbaum, “Optimal kinematic control of an autonomous underwater vehicle,” IEEE Trans. Automat. Control 54 (7), 1623–1626 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  18. A. V. Borisov and I. S. Mamaev, Dynamics of a Rigid Body. Hamiltonian Methods, Integrability, Chaos (IKI, Moscow–Izhevsk, 2005) [in Russian].

    MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, Russia

    A. V. Borisov

  2. Udmurt State University, Izhevsk, Russia

    A. V. Borisov, E. V. Vetchanin & A. A. Kilin

  3. Kalashnikov Udmurt State Technical University, Izhevsk, Russia

    E. V. Vetchanin

  4. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

    A. A. Kilin

Authors
  1. A. V. Borisov
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  2. E. V. Vetchanin
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  3. A. A. Kilin
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Corresponding author

Correspondence to A. V. Borisov.

Additional information

Original Russian Text © A. V. Borisov, E. V. Vetchanin, A. A. Kilin, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 4, pp. 503–513.

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Borisov, A.V., Vetchanin, E.V. & Kilin, A.A. Control of the motion of a triaxial ellipsoid in a fluid using rotors. Math Notes 102, 455–464 (2017). https://doi.org/10.1134/S0001434617090176

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  • DOI: https://doi.org/10.1134/S0001434617090176

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