Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CONTROLLABILITY OF ROLLING BODIES WITH REGULAR SURFACES

  • Moghadasi, S. Reza (Department of Mathematical Science Sharif University of Technology)
  • Received : 2014.05.14
  • Published : 2016.07.01
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Abstract

A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces.

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References

  1. A. Agrachev, Rolling balls and octonions, Proc. Steklov Inst. Math. 258 (2007), no. 1, 13-22. https://doi.org/10.1134/S0081543807030030
  2. A. Agrachev and Y. Sachkov, Control theory from the geometric viewpoint, Encyclopae-dia of Mathematical Sciences, 87, Control Theory and Optimization II, Springer-Verlag, Berlin, 2004.
  3. R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math. 114 (1993), no. 2, 435-461. https://doi.org/10.1007/BF01232676
  4. A. Chelouah and Y. Chitour, On the motion planning of rolling surfaces, Forum Math. 15 (2003), no. 5, 727-758. https://doi.org/10.1515/form.2003.039
  5. Y. Chitour and P. Kokkonen, Rolling of manifolds and controllability in dimension three, in Memoires de la Societe Mathematique de France, 2015.
  6. M. P. Do Carmo, Differential geometry of curves and surfaces, Prentice-Hall Englewood Cliffs, 1976.
  7. M. Godoy Molina, E. Grong, I. Markina, and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds, J. Dyn. Control Syst. 18 (2012), no. 2, 181-214. https://doi.org/10.1007/s10883-012-9139-2
  8. E. Grong, Controllability of rolling without twisting or slipping in higher dimensions, SIAM J. Control Optim. 50 (2012), no. 4, 2462-2485. https://doi.org/10.1137/110829581
  9. V. Jurdjevic, The geometry of the plate-ball problem, Archive for Rational Mechanics and Analysis 124 (1993), no. 4, 305-328. https://doi.org/10.1007/BF00375605
  10. V. Jurdjevic, Geometric Control Theory, Cambridge university press, 1997.
  11. W. Klingenberg, A Course in Differential Geometry, Springer Science & Business Media, 51, 2013.
  12. M. Levi, Geometric phases in the motion of rigid bodies, Arch. Rational Mech. Anal. 123 (1993), no. 3, 305-328. https://doi.org/10.1007/BF00375583
  13. Z. Li and J. Canny, Motion of two rigid bodies with rolling constraint, IEEE Trans. Automat. Control 6 (1990), no. 1, 62-72. https://doi.org/10.1109/70.88118
  14. A. Marigo and A. Bicchi, Rolling bodies with regular surface: Controllability theory and applications, IEEE Trans. Automat. Control 45 (2000), no. 9, 1586-1599. https://doi.org/10.1109/9.880610
  15. A. Marigo, M. Ceccarelli, S. Piccinocchi, and A. Bicchi, Planning motions of polyhedral parts by rolling, Algorithmica 26 (2000), no. 3-4, 560-576. https://doi.org/10.1007/s004539910024
  16. S. R. Moghadasi, Rolling of a body on a plane or a sphere: A geometric point of view, Bull. Austral. Math. Soc. 70 (2004), no. 2, 245-256. https://doi.org/10.1017/S0004972700034468
  17. J. Monforte, Geometric, control, and numerical aspects of nonholonomic systems, 1793, Springer Verlag, 2002.
  18. D. J. Montana, The kinematics of contact and grasp, Int. J. Rob. Res. 7 (1988), no. 3, 17-32. https://doi.org/10.1177/027836498800700302
  19. J. A. Zimmerman, Optimal control of the sphere $S^n$ rolling on $E^n$, Math. Control Signals Systems 17 (2005), no. 1, 14-37. https://doi.org/10.1007/s00498-004-0143-2