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.
Ideal fuid Controllability Kirchhoff equations Circulation around the body
Mathematics Subject Classification (2010)
70Q05 70Hxx 76Bxx
This is a preview of subscription content, log in to check access.
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.
Agrachev AA, Sachkov Y. Control theory from the geometric viewpoint: Springer Science & Business Media; 2004.Google Scholar
Arnold VI. Mathematical methods of classical mechanics: Springer Science & Business Media; 1989.Google Scholar
Bonnard B. Contrôlabilité des systèmes nonlinéaires. C R Acad Sci Paris Sér 1 1981;292:535–537.MATHGoogle Scholar
Bonnard B, trajectories Chyba M., Vol. 40. Their role in control theory Singular: Springer Science & Business Media; 2003.Google Scholar
Bolotin SV. The problem of optimal control of a Chaplygin ball by internal rotors. Rus J Nonlin Dyn 2012;8(4):837–852. Russian.MATHGoogle Scholar
Chow W, Über L. Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math Ann 1939;117(1):98–105.MathSciNetMATHGoogle Scholar
Chyba M, Leonard NE, Sontag ED. Optimality for underwater vehicles. IEEE 1998,2001;5:4204–4209.Google Scholar
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.MathSciNetCrossRefMATHGoogle Scholar
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.CrossRefMATHGoogle Scholar
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.MathSciNetCrossRefMATHGoogle Scholar
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.MathSciNetCrossRefMATHGoogle Scholar
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
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
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