Asymptotic perturbed feedback linearisation of underactuated Euler's dynamics

A unifying methodology is introduced for smooth asymptotic stabilisation of underactuated rigid body dynamics under one and two degrees of actuation. The methodology is based on the concept of generalised inversion, and it aims to realise a perturbation from the unrealisable feedback linearising transformation. A desired linear dynamics in a norm measure of the angular velocity components about the unactuated axes is evaluated along solution trajectories of Euler's underactuated dynamical equations resulting in a linear relation in the control variables. This relation is used to assess asymptotic stabilisability of underactuated rigid bodies with arbitrary values of inertia parameters, and generalised inversion of the relation produces a control law that consists of particular and auxiliary parts. The generalised inverse in the particular part is scaled by a dynamic factor such that it uniformly converges to the Moore-Penrose inverse, and the null-control vector in the auxiliary part is chosen for asymptotically stable perturbed feedback linearisation of the underactuated system.