The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page

Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem

Given a submersion $\pi:Q \to M$ with an Ehresmann connection $\mathcal{H}$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a generalization of the magnetic force. This generalized force is described using the curvature of $\mathcal{H}$ along with a new form of parallel transport of covectors vanishing on $\mathcal{H}$. Using the Pontryagin maximum principle, we apply this theory to optimal control problems $M$ and $Q$ to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length.Comment: 31 page

COMPLETE CONTROLLABILITY OF THE KINEMATIC EQUATIONS DESCRIBING PURE ROLLING OF GRASSMANNIANS

This paper studies the controllability properties of certain nonholo nomic control systems, describing the rolling motion of Grassmann manifolds over the affine tangent space at a point. The control functions correspond to the freedom of choosing the rolling curve. The nonholonomic constraints are imposed by the non-slip and non-twist conditions on the rolling. These systems are proved to be controllable in some submanifold of the group of isometries of the space where the two rolling manifolds are embedded. The constructive proof of controllability is also partially addressed.info:eu-repo/semantics/publishedVersio

The effect of lateral controls in producing motion of an airplane as computed from wind-tunnel data

This report presents the results of an analytical study of the lateral controllability of an airplane in which both the static rolling and yawing moments supplied by the controls and the reactions due to the inherent stability of the airplane have been taken into account. The investigation was undertaken partly for the purpose of coordinating the results of a long series of wind-tunnel investigations with phenomena observed in flight tests; for this reason a hypothetical average airplane, embodying the essential characteristics of both wind-tunnel models and the full-size test airplanes, was assumed for the study

Rolling Manifolds: Intrinsic Formulation and Controllability

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 no relative spin (or twist) of one manifold with respect to the other one. As for the rolling problem $(R)$, there is no relative spin and also no relative slip. Since the manifolds are not assumed to be embedded into an Euclidean space, we provide an intrinsic description of the two constraints "without spinning" and "without slipping" in terms of the Levi-Civita connections $\nabla^{g}$ and \nabla^{\hg}. For that purpose, we recast the two rolling problems within the framework of geometric control and associate to each of them a distribution and a control system. We then investigate the relationships between the two control systems and we address for both of them the issue of complete controllability. For the rolling $(NS)$, the reachable set (from any point) can be described exactly in terms of the holonomy groups of $(M,g)$ and (\hM,\hg) respectively, and thus we achieve a complete understanding of the controllability properties of the corresponding control system. As for the rolling $(R)$, the problem turns out to be more delicate. We first provide basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, we point out the role played by a curvature tensor defined on the state space, that we call the \emph{rolling curvature}. In the case where one of the manifolds is a space form (let say (\hM,\hg)), we show that it is enough to roll along loops of $(M,g)$ and the resulting orbits carry a structure of principal bundle which preserves the rolling $(R)$ distribution. In the zero curvature case, we deduce that the rolling $(R)$ is completely controllable if and only if the holonomy group of $(M,g)$ is equal to SO(n). In the nonzero curvature case, we prove that the structure group of the principal bundle can be realized as the holonomy group of a connection on $TM\oplus \R$, that we call the rolling connection. We also show, in the case of positive (constant) curvature, that if the rolling connection is reducible, then $(M,g)$ admits, as Riemannian covering, the unit sphere with the metric induced from the Euclidean metric of $\R^{n+1}$. When the two manifolds are three-dimensional, we provide a complete local characterization of the reachable sets when the two manifolds are three-dimensional and, in particular, we identify necessary and sufficient conditions for the existence of a non open orbit. Besides the trivial case where the manifolds $(M,g)$ and (\hM,\hg) are (locally) isometric, we show that (local) non controllability occurs if and only if $(M,g)$ and (\hM,\hg) are either warped products or contact manifolds with additional restrictions that we precisely describe. Finally, we extend the two types of rolling to the case where the manifolds have different dimensions

Investigation of all-wheel-drive off-road vehicle dynamics augmented by visco-lock devices

A peculiarity of AWD off-road vehicles is that their behaviour depends not only on the total power, provided by the engine, but also on its distribution among the drive axles/wheels. In turn, this distribution is largely regulated by the drivetrain layout and its torque distribution devices. At the output of the drivetrain system, the torque is constrained by the interaction between the wheels and the soft soil. For off-road automotive applications, the design of drivetrain systems has usually been largely dominated by the mobility requirements. With the growing demand to have a multipurpose on/off road vehicle with improved manoeuvrability over deformable soil, particularly at higher speed, the challenges confronting vehicle designers have become more complex. The thesis presents a novel integrated numerical approach to assess the dynamic behaviour of all-wheel-drive vehicles whilst operating over deformable soil terrain. [Continues.