Journal Article

Controllability of Rolling without Twisting or Slipping in Higher Dimensions

Grong, Erlend

2012

Overview

We describe how the kinematic system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\\widehat M$ without twisting or slipping can be lifted to a nonholonomic system defined on the product of the oriented orthonormal frame bundles belonging to the two manifolds. By using known properties of forms known as Cartan's moving frame, we obtain sufficient conditions for the local controllability of the system in terms of the curvature tensors and the sectional curvatures of the manifolds involved. By using the information from these calculations, we show that we need only consider normal extremals, when looking for a rolling of minimal length, connecting two given configurations. We also give some results for controllability in the particular cases when $M$ and $\\widehat M$ are locally symmetric or complete. [PUBLICATION ABSTRACT]

• Publisher: Society for Industrial and Applied Mathematics
• Subject: Bundles / Controllability / Curvature / Euclidean space / Frames / Geometry / Kinematics / Manifolds / Mathematical analysis / Optimization / Twisting