Nonholonomic systems are, so to speak, mechanical systems with a prescribed restriction on the velocities. A virtual nonholonomic constraint is a controlled invariant distribution associated with an affine connection mechanical control system. A Riemannian homogeneous space is, a Riemannian manifold that looks the same everywhere, as you move through it by the action of a Lie group. These Riemannian manifolds are not necessarily Lie groups themselves, but nonetheless possess certain symmetries and invariances that allow for similar results to be obtained. In this work, we introduce the notion of virtual constraint on Riemannian homogeneous spaces in a geometric framework which is a generalization of the classical controlled invariant distribution setting and we show the existence and uniqueness of a control law preserving the invariant distribution. Moreover we characterize the closed-loop dynamics obtained using the unique control law in terms of an affine connection. We illustrate the theory with new examples of nonholonomic control systems inspired by robotics applications.

In order to increase the number of situations in which an intelligent vehicle can operate without human intervention, lateral control is required to accurately guide it in a reference trajectory regardless of the shape of the road or the longitudinal speed. Some studies address this problem by tuning a controller for low and high speeds and including an output adaptation law. In this paper, a strategy framed in the Model-Free Control paradigm is presented to laterally control the vehicle over a wide speed range. Tracking quality, system stability and passenger comfort are thoroughly analyzed and compared to similar control structures. The results obtained both in simulation and with a real vehicle show that the developed strategy tracks a large number of trajectories with high degree of accuracy, safety and comfort.