JOURNAL BROWSE
Search
Advanced SearchSearch Tips
CONTROLLABILITY OF ROLLING BODIES WITH REGULAR SURFACES
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
CONTROLLABILITY OF ROLLING BODIES WITH REGULAR SURFACES
Moghadasi, S. Reza;
  PDF(new window)
 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.
 Keywords
controllability;rolling bodies;geodesic curvature;isometry group;
 Language
English
 Cited by
 References
1.
A. Agrachev, Rolling balls and octonions, Proc. Steklov Inst. Math. 258 (2007), no. 1, 13-22. crossref(new window)

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. crossref(new window)

4.
A. Chelouah and Y. Chitour, On the motion planning of rolling surfaces, Forum Math. 15 (2003), no. 5, 727-758.

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. crossref(new window)

8.
E. Grong, Controllability of rolling without twisting or slipping in higher dimensions, SIAM J. Control Optim. 50 (2012), no. 4, 2462-2485. crossref(new window)

9.
V. Jurdjevic, The geometry of the plate-ball problem, Archive for Rational Mechanics and Analysis 124 (1993), no. 4, 305-328. crossref(new window)

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. crossref(new window)

13.
Z. Li and J. Canny, Motion of two rigid bodies with rolling constraint, IEEE Trans. Automat. Control 6 (1990), no. 1, 62-72. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)