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.