Finite-time stable trajectory tracking and pointing control for a class of underactuated vehicles in SE(3)

In this paper, an integrated guidance and control scheme for finite-time position and pointing direction tracking of a class of underactuated vehicles is presented. The underactuated vehicle is considered to be a rigid body with translational control applied only along a fixed body frame direction and rotational control that can be applied along all three body-fixed coordinate axes. The combined rotational and translational motion of the vehicle has six degrees of freedom and only four degrees of freedom actuated; thus the system as a whole is underactuated. This model is representative of a number of commonly used aerial vehicles including quadrotors, vertical take-off and landing (VTOL) aircraft, fixed-wing aerial vehicles and spacecraft. The guidance and control problem is formulated in the space of the special Euclidean group of rigid body motions, SE(3), using the framework of geometric mechanics. The proposed control ensures that the difference between the actual and the desired position trajectory is stabilized to zero in finite time. Additionally, through the proposed feedback control, the specified pointing vector in the body-fixed frame of the vehicle tracks the desired pointing direction in finite time. Almost global finite-time stability of the overall closed-loop system over the state space is shown analytically and illustrated through numerical simulations.