  This paper addresses the problem of designing a trajectory tracking control
law for a quadrotor UAV, subsequent to complete failure of a single rotor. The
control design problem considers the reduced state space which excludes the
angular velocity and orientation about the vertical body axis. The proposed
controller enables the quadrotor to track the orientation of this axis, and
consequently any prescribed position trajectory using only three rotors. The
control design is carried out in two stages. First, in order to track the
reduced attitude dynamics, a geometric controller with two input torques is
designed on the Lie-Group $SO(3)$. This is then extended to $SE(3)$ by
designing a saturation based feedback law, in order to track the center of mass
position with bounded thrust. The control law for the complete dynamics
achieves exponential tracking for all initial conditions lying in an open-dense
subset. The novelty of the geometric control design is in its ability to
effectively execute aggressive, global maneuvers despite complete loss of a
rotor. Numerical simulations on models of a variable pitch and a conventional
quadrotor have been presented to demonstrate the practical applicability of the
control design.
