Symmetries of the rolling model

Yacine Chitour (1) , M.-G. Molina , Petri Kokkonen Kokkonen

1 L2S - Laboratoire des signaux et systèmes

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds (M, g) and ({\hat{M}},\hat{g}) rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle projection from the state space Q of the rolling model onto M is a principal bundle if and only if {\hat{M}} has constant sectional curvature. Additionally, we prove that when M and {\hat{M}} have different constant sectional curvatures and dimension n\ge 3, the rolling distribution is never flat, contrary to the two dimensional situation of rolling two spheres of radii in the proportion 1{:}3, which is a well-known system satisfying É. Cartan’s flatness condition.

Domains: Automatic Control Engineering; Automatic

Yacine Chitour, M.-G. Molina, Petri Kokkonen Kokkonen. Symmetries of the rolling model. Mathematische Zeitschrift, 2015, 281 (4), pp.783-805. ⟨10.1007/s00209-015-1508-6⟩. ⟨hal-01271285⟩