Rolling Manifolds of Different Dimensions

Abstract

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.

Appendix

In this section we briefly show how one writes the control system Σ _(R) in local orthonormal frames.

Let (F _i )_1≤i≤n and $(\hat{F}_{j})_{1\leq j\leq\hat{n}}$ be local oriented orthonormal frames on M and $\hat{M}$ respectively and let $q_{0}=(x_{0},\hat{x}_{0};A_{0})\in Q$ such that x ₀, $\hat{x}_{0}$ belong to the domains of definition V and $\hat{V}$ of the frames. Let $q(t)=(\gamma(t),\hat{\gamma}(t);A(t))$, t∈[0,1], be a curve in Q so that γ⊂V and $\hat{\gamma}\subset\hat{V}$. For every t∈[0,1], define the unique element $\mathcal{R}(t)$ in $\mathrm{\mathit{SO}}(n,\hat{n})$ verifying

$$\bigl(A(t)F_1|_{\gamma(t)},\dots,A(t)F_n|_{\gamma(t)} \bigr) = (\hat{F}_1|_{\hat{\gamma}(t)},\dots,\hat{F}_{\hat {n}}|_{\hat{\gamma}(t)} )\mathcal{R}(t) $$

Define Christoffel symbols $\varGamma\in T^{*}_{x} M\otimes\mathfrak {so}(n)$ and $\hat{\varGamma}\in T^{*}_{\hat{x}} \hat{M}\otimes \mathfrak{so}(\hat{n})$ by

$$\varGamma(X)_i^l=g(\nabla_X F_i, F_l),\quad \quad \hat{\varGamma}(\hat{X})_j^k= \hat{g}(\hat{\nabla}_{\hat{X}} \hat {F}_j, \hat{F}_k), $$

with 1≤i,k≤n, $1\leq j,k\leq\hat{n}$ and X∈T _x M, $\hat{X}\in T_{\hat{x}}\hat{M}$.

There are unique measurable functions $u^{i}:[0,1]\to\mathbb{R}$, 1≤i≤n, such that, for a.e. t∈[0,1],

$$\dot{\gamma}(t)= (F_1|_{\gamma(t)},\dots,F_n|_{\gamma(t)} )\left ( \begin{array}{c}u^1(t)\\ \vdots\\ u^n(t) \end{array} \right ). $$

As one can easily verify, the conditions of no-slip (7) and no-spin (6) translate for $(\hat{\gamma}(t),\mathcal{R}(t))\in\hat{M}\times \mathrm{\mathit{SO}}(n)$ precisely to

$$\begin{aligned} & \mbox{(no-slip)}\quad \dot{\hat{\gamma}}(t)= (\hat {F}_1|_{\hat{\gamma}(t)}, \dots,\hat{F}_{\hat{n}}|_{\hat{\gamma }(t)} )\mathcal{R}(t)\left ( \begin{array}{c}u^1(t)\\ \vdots\\ u^n(t) \end{array} \right ), \\ &\mbox{(no-spin)}\quad \dot{\mathcal{R}}(t)=\mathcal{R}(t)\varGamma \bigl( \dot{\gamma}(t) \bigr)-\hat{\varGamma} \bigl(\dot{\hat{\gamma}}(t) \bigr) \mathcal{R}(t), \end{aligned}$$

for a.e. t∈[0,1]. Moreover, the latter no-spin condition can also be written as

$$\dot{\mathcal{R}}(t)=\sum_{i=1}^n u^i(t) \Biggl(\mathcal{R}(t)\varGamma (F_i|_{\gamma(t)})- \sum_{j=1}^{\hat{n}} \mathcal{R}_{ji}(t) \hat {\varGamma}(\hat{F}_j|_{\hat{\gamma}(t)})\mathcal{R}(t) \Biggr), $$

for a.e. t∈[0,1], where $\mathcal{R}_{ji}(t)$ is the element at j-th row, i-th column of $\mathcal{R}(t)$. From this local form, one clearly sees that the rolling system Σ _R is a driftless control affine system (see [2, 11] for more details on control systems).