# Control of Body Motion in an Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation Around the Body

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## Abstract

In this paper, we study the controlled motion of an arbitrary two-dimensional body in an ideal fluid with a moving internal mass and an internal rotor in the presence of constant circulation around the body. We show that by changing the position of the internal mass and by rotating the rotor, the body can be made to move to a given point and discuss the influence of nonzero circulation on the motion control. We have found that in the presence of circulation around the body the system cannot be completely stabilized at an arbitrary point of space, but fairly simple controls can be constructed to ensure that the body moves near the given point.

## Keywords

Ideal fuid Controllability Kirchhoff equations Circulation around the body## Mathematics Subject Classification (2010)

70Q05 70Hxx 76Bxx## Notes

### Acknowledgments

The authors thank A.V. Borisov and I.S. Mamaev for fruitful discussions. The work of E.V. Vetchanin was supported by the RFBR grant 15-08-09093-a. The work of A.A. Kilin was supported by the RFBR grant 14-01-00395-a.

## References

- 1.Agrachev AA, Sachkov Y. Control theory from the geometric viewpoint: Springer Science & Business Media; 2004.Google Scholar
- 2.Arnold VI. Mathematical methods of classical mechanics: Springer Science & Business Media; 1989.Google Scholar
- 3.Bonnard B. Contrôlabilité des systèmes nonlinéaires. C R Acad Sci Paris Sér 1 1981;292:535–537.zbMATHGoogle Scholar
- 4.Bonnard B, trajectories Chyba M., Vol. 40. Their role in control theory Singular: Springer Science & Business Media; 2003.Google Scholar
- 5.Bolotin SV. The problem of optimal control of a Chaplygin ball by internal rotors. Rus J Nonlin Dyn 2012;8(4):837–852. Russian.zbMATHGoogle Scholar
- 6.Borisov AV, Mamaev IS. On the motion of a heavy rigid body in an ideal fluid with circulation. CHAOS 2006;16(1):013118. (7 pages).MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Borisov AV, Kilin AA, Mamaev IS. How to control chaplygin’s sphere using rotors. Regul Chaotic Dyn 2012;17(3–4):258–272.MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Borisov AV, Kilin AA, Mamaev IS. How to control the chaplygin ball using rotors. II. Regul Chaotic Dyn 2013;18(1–2):144–158.MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Borisov AV, Kilin AA, Mamaev IS. Dynamics and Control of an Omniwheel Vehicle. Regul Chaotic Dyn 2015;20(2):153–172.MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Borisov AV, Kozlov VV, Mamaev IS. Asymptotic stability and associated problems of failing rigid body. Regul Chaotic Dyn 2007;12(5):531–565.MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Borisov AV, Mamaev IS, Kilin AA, Bizyaev IA. Qualitative Analysis of the Dynamics of a Wheeled Vehicle. Regul Chaotic Dyn 2015;20(6):739–751.MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Chaplygin S. On the influence of a plane-parallel flow of air on moving through it a cylindrical wing. Tr Cent Aerohydr Inst 1926;19:300–382.Google Scholar
- 13.Childress S, Spagnolie SE, Tokieda T. A bug on a raft: recoil locomotion in a viscous fluid. J Fluid Mech 2011;669:527–556.MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Chow W, Über L. Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math Ann 1939;117(1):98–105.MathSciNetzbMATHGoogle Scholar
- 15.Chyba M, Leonard NE, Sontag ED. Optimality for underwater vehicles. IEEE 1998,2001;5:4204–4209.Google Scholar
- 16.Chyba M, Leonard NE, Sontag ED. Singular trajectories in multi-input time-optimal problems: application to controlled mechanical systems. J Dyn Control Syst 2003;9(1):103–129.MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Crouch PE. Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models. IEEE Trans Autom Control 1984;29(4):321–331.CrossRefzbMATHGoogle Scholar
- 18.Ivanov AP. On the control of a robot ball using two omniwheels. Regul Chaotic Dyn 2015;20(4):441–448.MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Jurdjevic V. Geometric control theory. Cambridge: University Press; 1997.zbMATHGoogle Scholar
- 20.Karavaev YL, Kilin AA. The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform. Regul Chaotic Dyn 2015;20(2):134–152.MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Kilin AA, Vetchanin EV. The contol of the motion through an ideal fluid of a rigid body by means of two moving masses. Nelin Dinam 2015;11(4):633–645. (in Russian).CrossRefzbMATHGoogle Scholar
- 22.Kirchhoff G., Hensel K. Vorlesungen über mathematische Physik. Mechanik. Leipzig: BG Teubner; 1874, p. 489.Google Scholar
- 23.Kozlov VV, Onishchenko DA. The motion in a perfect fluid of a body containing a moving point mass. J Appl Math Mech 2003;67(4):553–564.MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Kozlov VV, Ramodanov S. On the motion of a variable body through an ideal fluid. PMM 2001;65(Vyp. 4):529–601.Google Scholar
- 25.Lamb H. Hydrodynamics. New York: Dover; 1945, p. 728.Google Scholar
- 26.Leonard NE, Marsden JE. Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry. Phys D: Nonlinear Phenom 1997;105 (1):130–162.MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Leonard NE. Stability of a bottom-heavy underwater vehicle. Automatica 1997; 33(3):331–346.MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Murray RM, Sastry SS. Nonholonomic motion planning: steering using sinusoids. IEEE Trans Autom Control 1993;38(5):700–716.MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Ramodanov SM, Tenenev VA, Treschev DV. Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid. Regul Chaotic Dyn 2012;17 (6):547–558.MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Rashevskii PK. About connecting two points of complete non-holonomic space by admissible curve (in Russian). Uch Zapiski Ped Inst Libknexta;2:83–94.Google Scholar
- 31.Steklov VA. On the motion of a rigid body through a fluid. Article 1. Soob Khark Mat Obsch 1891;2(5–6):C.209–235.Google Scholar
- 32.Svinin M, Morinaga A, Yamamoto M. On the dynamic model and motion planning for a class of spherical rolling robots. IEEE Int Conf Robot Autom 2012;2012: 3226–3231.Google Scholar
- 33.Vankerschaver J, Kanso E, Marsden JE. The dynamics of a rigid body in potential flow with circulation. Regul Chaotic Dyn 2010;15(4-5):606–629.MathSciNetCrossRefzbMATHGoogle Scholar
- 34.Vetchanin EV, Kilin AA. Free and controlled motion of a body with moving internal mass though a fluid in the presence of circulation around the body. Dokl Phys 2016;466(3):293–297.Google Scholar
- 35.Vetchanin EV, Mamaev IS, Tenenev VA. The self-propulsion of a body with moving internal masses in a viscous fluid. Regul Chaotic Dyn 2013;18(1-2):100–117.MathSciNetCrossRefzbMATHGoogle Scholar
- 36.Woolsey CA, Leonard NE. Stabilizing underwater vehicle motion using internal rotors. Automatica 2002;38(12):2053–2062.MathSciNetCrossRefzbMATHGoogle Scholar