This thesis studies geometric formulations of a slow-fast realization of nonholonomic mechanical control systems mediated by strong friction forces. Such a slow-fast realization is motivated by wheeled mobile robotic (WMR) systems and viewed as a singular perturbation of nonholonomic dynamics. For sufficiently strong friction forces, the dynamics emit an attractive normally hyperbolic invariant manifold interpreted as the perturbation of ideal nonholonomic constraints. In the context of wheeled robotic systems, this invariant manifold describes small slip velocities at the wheel contact points. A Hamiltonian slow-fast realization is formulated with Poisson geometry and a Lagrangian realization is modelled as a forced affine connection system.

On the Lagrangian side, we propose a coordinate-free procedure to decompose the dynamics into slow and fast directions using an affine connection approach. The invariant manifold – describing the slip-velocities – is represented by the image of a section expanded as a power series. We propose a novel invariance condition and a novel coordinate-free recursive procedure to approximate the invariant manifold. We provide first- and second-order approximations of the invariant manifold and consider the zeroth- and first-order equations of motion. We consider a differential robot as a case study.

On the Hamiltonian side, we propose a systematic decomposition of the control system into slow and fast directions using the kinetic energy metric and the geometry of fast friction reaction forces. The invariant manifold associated with the singularly perturbed system is approximated with a power series, which can be recursively computed based on an invariance condition. Accordingly, we develop a novel recursive procedure to input-output linearize the slow subsystem (restricted system to the invariant manifold) by dynamic-state feedback transformations. Using this procedure, we introduce the notion of almost input-output linearization for the approximated slow subsystem and design a trajectory-tracking PD controller to compensate for the violations of the nonholonomic constraints.

Closed-loop stability analysis is performed on both the slow subsystem and the unrestricted system, under the almost input-output linearizing recursive control law. We prove that if the internal error dynamics of the nonholonomic system are exponentially stable at the origin, then the internal error dynamics of the slow subsystem remain exponentially stable under some explained conditions. Using this result, we prove that the output error dynamics for the closed-loop (unrestricted) system subject to the proposed recursive control law remains uniformly bounded at the origin, while for the full power series expansion we have asymptotic stability. Our approach is illustrated through a numerical case study on a differential robot.