Almost global attitude stabilisation of an underactuated axially symmetric 3-D pendulum

We investigate the problem of stabilising the attitude of an axially symmetric 3-D pendulum. The system is assumed to be actuated by two torques acting orthogonally to the symmetry axis. In the process of stabilising the system, two main difficulties are encountered: first, the presence of a well-known topological obstruction to the global stabilisation by means of continuous feedback laws, and second, the fact that the linearised system around the target equilibrium is not controllable due to the strong underactuation of the system. To cope with the first issue, in this paper we look for feedback laws capable of stabilising the system starting from almost all initial conditions. The second issue entails the impossibility of stabilising the system exponentially by means of a smooth feedback. The contributions of this paper are (1) to propose a family of smooth feedback laws capable of almost-globally stabilising the system with a polynomial rate of convergence, and (2) to define a second family of feedback laws presenting a discontinuity at the target attitude configuration and continuous elsewhere inducing almost global exponential stabilisation.