Recursive Input-Output Linearization for Slow-Fast Realization of Nonholonomic Hamiltonian Control Systems

In this paper, we study a slow-fast realization of nonholonomic Hamiltonian control systems mediated by strong friction forces which is viewed as a singular perturbation of the nonholonomic system. We propose a systematic decomposition of the perturbed dynamics into slow and fast directions using the kinetic energy metric and the geometry of friction forces. The (slow) invariant manifold is identified by a set of invariance conditions resulting in partial differential equations that generally do not have an analytic solution. We approximate the invariant manifold along with the control inputs with power series and show that using this approximation the invariance conditions admit an inherent recursion. Accordingly, we develop a recursive procedure to perform dynamic input-output linearization of the approximated slow system. We consider the output trajectory tracking problem using a PD control law on the
$N^{th}$
order approximation of the invariant manifold. Closed-loop stability analysis is performed on both the invariant manifold and the full phase space of the system. We prove that if the internal dynamics of the nonholonomic system is exponentially stable, then the perturbed system remains asymptotically stable. Moreover, we prove that the output error dynamics is uniformly bounded when applying the approximated control law, with bounds dependent on the control gains and strength of the friction force. Our approach is illustrated through a numerical case study on a differential robot.