rolling manifolds

Geometry Analysis and Dynamics on sub-Riemannian manifolds

**9:00-9:15**Welcome words**9:15-10:00**Irina Markina*(Department of Mathematics, University of Bergen)*

Title:*Rolling of a manifold M over a sphere: controllability, holonomy and geometry of M*

Abstract: We state the intrinsic problem of rolling of smooth n-dimensional Riemannian manifolds over each other without twisting and slipping. We describe the non-holonomic distribution defined by the kinematic constraints of no twisting and no slipping and study the controllability of the rolling system. If one of the manifolds is n-dimensional unit sphere, then the controllability problem is closely related to the holonomy group of a metric cone over the other manifold. From the other side the holonomy group of a metic cone over a manifold M decides the geometry of the manifold M itself.**10:00-10:300**Coffee break**10:30-11:15**Petri Kokkonen*(Varian Medial Systems Finland Oy)*

Title:*Rolling Cartan Geometries*

Abstract: A model for generalized rolling in the context of arbitrary Cartan geometries will be presented and studied. As an application, we will briefly discuss the rolling of a pseudo-Riemannian manifold against the corresponding constant curvature spaces.**11:20-12:05**Amina Mortada*(LSS, Univesité Paris-Sud)*

Title:*Horizontal holonomy*

Abstract: In this talk, we consider a smooth connected finite-dimensional manifold M, a complete affine connection D with holonomy group H and a smooth distribution Δ. We define H_{Δ}as the subgroup of H obtained by transporting frames D-parallely only along loops tangent to Δ. Assuming that Δ is completely controllable, the question we address then is the following: is, in general, (the closure of) H_{Δ}equal to H? The answer is no by means of an example.

**14:30-15:15**Erlend Grong*(University of Luxembourg)*

Title:*Optimal solutions to the rolling problem*

Abstract: We look at the problem of rolling two Riemannian manifolds from one configuration to another along a curve of shortest possible length. In order to avoid complicated calculations, we introduce some general tools for solving optimal control problems on submersions. We end by discussing optimal solutions of the rolling problem and its relation to Riemannian elastica and gauge theory.**15:15-15:45**Coffee break**15:45-16:30**Fátima Silva Leite*(Department of Mathematics, University of Coimbra)*

Title:*Solving interpolation problems on manifolds using rolling motions*

Abstract: Classical methods to smoothly interpolate time-labelled data in Euclidean spaces have been generalized since late 80's to deal with data belonging to manifolds. The main drawback of these methods is that explicit solutions are very hard to find. In this talk, we will show how rolling motions can be used successfully to generate interpolating curves given in closed form. This will be illustrated for ellipsoids.**16:35-17:20**Mikhail Svinin*(Department of Mechanical Engineering, Kyushu University)*

Title:*Motion planning algorithms for spherical rolling robots*

Abstract: In this talk we address the motion planning problem for self-propelling spherical robots rolling on a plane. First, we first discuss kinematic planning algorithms, based on design of the contact curves, for the pure rolling case. Two dynamic models are then introduced, one for the robot actuated by internal rotors and one for that actuated by pendulum. The dynamic models admit twisting, and the kinematic algorithms need to be modified. A motion planning strategy is finally constructed and illustrated by simulation examples.

Yacine Chitour *(LSS, Univesité Paris-Sud)* and Mauricio Godoy Molina *(Department of Mathematics, University of Bergen)*