Section: New Results

New results: geometric control

A first set of new results concerns sub-Riemannian geometry.

• In [3] we continued the study of almost-Riemannian structures, which are rank-varying sub-Riemannian structures locally generated by a number of vector fields equal to the dimension of the ambient manifold. In particular, two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which local orthonormal frames are Lie bracket generating pair of vector fields that can become collinear. We considered the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and studied the problem of Lipschitz equivalence between two such distances on a given compact oriented surface. We analyzed the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

• In [1] we studied nilpotent 2-step, corank 2 sub-Riemannian metrics. Such metrics naturally appear as nilpotent approximations of general sub-Riemannian ones. We exhibited optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we proved some smoothness properties of the spherical Hausdorff measure in the case of a generic 6-dimensional, 2-step corank 2 sub-Riemannian metric.

• In [12] we started from the remark that in Carnot–Carathéodory spaces the class of 1-rectifiable sets does not contain smooth non-horizontal curves. We were looking for a new definition of rectifiable sets including non-horizontal curves. We introduced, for any metric space, a new class of curves, called continuously metric differentiable of degree $k$, which are Hölder but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we defined $({\mathscr{H}}^k,1)$-rectifiable sets and showed a density result generalizing the corresponding one in Euclidean geometry. This theorem has been obtained as a consequence of computations of Hausdorff measures along curves, for which we gave an integral formula. In particular, we showed that both spherical and standard Hausdorff measures along curves coincide with a class of dimensioned lengths and are related with an interpolation complexity, for which estimates have already been obtained in Carnot–Carathéodory spaces.

A class of problems for which tracking and motion planning is crucial, is given by the control of unmanned aerial vehicles (UAV). In order to develop improved planning tasks that take into account payload requirements, optimal costs and obstacles avoidance (or no flight zones), it is important to develop reliable and flexible simulators. One such simulator for a UAV ground control station is proposed in [9] . The research focuses on the connection between the UAV trajectories and its sensors. Our proposal includes a module-based description of the architecture of the simulator and is based on a nonlinear model of a fixed wing aircraft.