Related papers: The rolling problem: overview and challenges

The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers…

We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that up to technical hypotheses, a…

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is…

In the present work we define the rolling of one pseudo-Riemannian manifold over another without slipping and twisting. We compare the definition of the rolling without slipping and twisting of two manifolds isometrically embedded into a…

We describe how the dynamical system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\hat M$ without twisting or slipping, can be lifted to a nonholonomic system of elements in the product of the oriented…

If $(M,g)$ and $(\hM,\hg)$ are two smooth connected complete oriented Riemannian manifolds of dimensions $n$ and $\hn$ respectively, we model the rolling of $(M,g)$ onto $(\hM,\hg)$ as a driftless control affine systems describing two…

We present an intrinsic formulation of the kinematic problem of two $n-$dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an $\frac{n(n+3)}2-$dimensional…

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modelled by a…

In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is…

We consider the problem of a sphere rolling of a curved surface and solve it by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be of pedagogical use in discussing both…

This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks,…

We discuss the rolling, without slip and without twist, of Stiefel manifolds equipped with $\alpha$-metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely by investigating…

We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.

This article presents an overview of robot learning and adaptive control applications that can benefit from a joint use of Riemannian geometry and probabilistic representations. The roles of Riemannian manifolds, geodesics and parallel…

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…

Riemannian geometry is a mathematical field which has been the cornerstone of revolutionary scientific discoveries such as the theory of general relativity. Despite early uses in robot design and recent applications for exploiting data with…

We generalize the classical Blaschke Rolling Theorem to convex domains in Riemannian manifolds of bounded sectional curvature and arbitrary dimension. Our results are sharp and, in this sharp form, are new even in the model spaces of…

A well-known and interesting family of sub-Riemannian space are the systems involving two balls rolling against each other without slipping or twisting. In this note, we show how the sub-Riemannian geodesics of these space, when the two…

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…