Time-Varying Feedback for Rigid Body Attitude Control

Stable attitude control of unmanned or autonomous operations of vehicles moving in three spatial dimensions is essential for safe and reliable operations. Rigid body attitude control is inherently a nonlinear control problem, as the Lie group of rigid body rotations is a compact manifold and not a linear (vector) space. Prior research has shown that the largest possible domain of convergence is provided by smooth attitude feedback control laws are obtained using a Morse function on SO(3) as a measure of the attitude stabilization or tracking error. A polar Morse function on SO(3) has four critical points, which precludes the possibility of global convergence of the attitude state. When used as part of a Lyapunov function on the state space (the tangent bundle TSO(3)) of attitude and angular velocity, it gives a globally continuous state-dependent feedback control scheme with the minimum of the Morse function as the almost globally asymptotically stable (AGAS) attitude state. In this work, we explore the use of explicitly time-varying gains for Morse functions for rigid body attitude control. This strategy leads to discrete switching of the indices of the three non-minimum critical points that correspond to the unstable equilibria of the feedback system. The resulting time-varying feedback controller is proved to be AGAS, with the additional desirable property that the time-varying gains destabilize the (locally) stable manifolds of these unstable equilibria. Numerical simulations of the feedback system with appropriate time-varying gains show that a trajectory starting from an initial state close to the stable manifold of an unstable equilibrium, converges to the desired stable equilibrium faster than the corresponding feedback system with constant gains.