Rolling Manifolds of Different Dimensions
Tóm tắt
If (M,g) and
$(\hat{M},\hat{g})$
are two smooth connected complete oriented Riemannian manifolds of dimensions n and
$\hat{n}$
respectively, we model the rolling of (M,g) onto
$(\hat{M},\hat{g})$
as a driftless control affine systems describing two possible constraints of motion: the first rolling motion (Σ)
NS
captures the no-spinning condition only and the second rolling motion (Σ)
R
corresponds to rolling without spinning nor slipping. Two distributions of dimensions
$(n + \hat{n})$
and n are then associated to the rolling motions (Σ)
NS
and (Σ)
R
respectively. This generalizes the rolling problems considered in Chitour and Kokkonen (Rolling manifolds and controllability: the 3D case, 2012) where both manifolds had the same dimension. The controllability issue is then addressed for both (Σ)
NS
and (Σ)
R
and completely solved for (Σ)
NS
. As regards to (Σ)
R
, basic properties for the reachable sets are provided as well as the complete study of the case
$(n,\hat{n})=(3,2)$
and some sufficient conditions for non-controllability.
Tài liệu tham khảo
Agrachev, A., Sachkov, Y.: An intrinsic approach to the control of rolling bodies. In: Proceedings of the CDC, Phoenix, vol. I, pp. 431–435 (1999)
Agrachev, A., Sachkov, Y.: Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences, vol. 87. Control Theory and Optimization, II. Springer, Berlin (2004)
Alouges, F., Chitour, Y., Long, R.: A motion planning algorithm for the rolling-body problem. IEEE Trans. Robot. 26(5), 827–836 (2010)
Bryant, R., Hsu, L.: Rigidity of integral curves of rank 2 distributions. Invent. Math. 114(2), 435–461 (1993)
Chelouah, A., Chitour, Y.: On the controllability and trajectories generation of rolling surfaces. Forum Math. 15, 727–758 (2003)
Chitour, Y., Godoy Molina, M., Kokkonen, P.: The rolling problem: overview and challenges (2013). arXiv:1301.6370
Chitour, Y., Godoy Molina, M., Kokkonen, P.: Symmetries of the rolling model (2013). arXiv:1301.2579
Chitour, Y., Kokkonen, P.: Rolling manifolds: intrinsic formulation and controllability. Preprint (2011). arXiv:1011.2925v2
Chitour, Y., Kokkonen, P.: Rolling manifolds and controllability: the 3D case. Submitted (2012)
Jean, F.: Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning. Springer, Berlin (2014)
Jurdjevic, V.: Geometric Control Theory. Cambridge Studies in Advanced Mathematics, vol. 52. Cambridge University Press, Cambridge (1997)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Wiley-Interscience, New York (1996)
Kokkonen, P.: A characterization of isometries between Riemannian manifolds by using development along geodesic triangles. Arch. Math. 48(3), 207–231 (2012)
Kokkonen, P.: Étude du modèle des variétés roulantes et de sa commandabilité. Ph.D. Thesis (2012). http://tel.archives-ouvertes.fr/tel-00764158
Marigo, A., Bicchi, A.: Rolling bodies with regular surface: controllability theory and applications. IEEE Trans. Autom. Control 45(9), 1586–1599 (2000)
Marigo, A., Bicchi, A.: Planning motions of polyhedral parts by rolling, algorithmic foundations of robotics. Algorithmica 26(3–4), 560–576 (2000)
Molina, M., Grong, E., Markina, I., Leite, F.: An intrinsic formulation of the problem of rolling manifolds. J. Dyn. Control Syst. 18(2), 181–214 (2012)
Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. Am. Math. Soc., Providence (1996)
Sharpe, R.W.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. Graduate Texts in Mathematics, vol. 166. Springer, New York (1997)
Vilms, J.: Totally geodesic maps. J. Differ. Geom. 4, 73–79 (1970)